Expression of type Equals

from the theory of proveit.linear_algebra.inner_products

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import H, K, x, y
from proveit.linear_algebra import InnerProd, InnerProdSpaces, VecSpaces
from proveit.logic import And, Equals, Forall, InClass, InSet

# build up the expression from sub-expressions
expr = Equals(InClass(H, InnerProdSpaces(K)), And(InClass(H, VecSpaces(K)), Forall(instance_param_or_params = [x, y], instance_expr = InSet(InnerProd(x, y), K))))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)\right) = \left(\left(H \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right) \land \left[\forall_{x, y}~\left(\left \langle x, y\right \rangle \in K\right)\right]\right)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 12 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	16, 8
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 11 operand: 25
9	Operation	operator: 12 operands: 13
10	Operation	operator: 14 operand: 18
11	Literal
12	Literal
13	ExprTuple	16, 17
14	Literal
15	ExprTuple	18
16	Variable
17	Operation	operator: 19 operand: 25
18	Lambda	parameters: 27 body: 21
19	Literal
20	ExprTuple	25
21	Operation	operator: 22 operands: 23
22	Literal
23	ExprTuple	24, 25
24	Operation	operator: 26 operands: 27
25	Variable
26	Literal
27	ExprTuple	28, 29
28	Variable
29	Variable