let o1, o2 be BinOp of (Subspaces V); ( ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
o1 . (A1,A2) = W1 + W2 ) & ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
o2 . (A1,A2) = W1 + W2 ) implies o1 = o2 )
assume A2:
for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
o1 . (A1,A2) = W1 + W2
; ( ex A1, A2 being Element of Subspaces V ex W1, W2 being Subspace of V st
( A1 = W1 & A2 = W2 & not o2 . (A1,A2) = W1 + W2 ) or o1 = o2 )
assume A3:
for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
o2 . (A1,A2) = W1 + W2
; o1 = o2
now for x, y being set st x in Subspaces V & y in Subspaces V holds
o1 . (x,y) = o2 . (x,y)let x,
y be
set ;
( x in Subspaces V & y in Subspaces V implies o1 . (x,y) = o2 . (x,y) )assume A4:
(
x in Subspaces V &
y in Subspaces V )
;
o1 . (x,y) = o2 . (x,y)then reconsider A =
x,
B =
y as
Element of
Subspaces V ;
reconsider W1 =
x,
W2 =
y as
Subspace of
V by A4, Def3;
o1 . (
A,
B)
= W1 + W2
by A2;
hence
o1 . (
x,
y)
= o2 . (
x,
y)
by A3;
verum end;
hence
o1 = o2
by BINOP_1:1; verum