let f1, f2 be BinOp of [.0,1.]; :: thesis: ( ( for a, b being Element of [.0,1.] holds
( ( a = b & b = 1 implies f1 . (a,b) = 1 ) & ( ( a <> 1 or b <> 1 ) implies f1 . (a,b) = ((a + b) - ((2 * a) * b)) / (1 - (a * b)) ) ) ) & ( for a, b being Element of [.0,1.] holds
( ( a = b & b = 1 implies f2 . (a,b) = 1 ) & ( ( a <> 1 or b <> 1 ) implies f2 . (a,b) = ((a + b) - ((2 * a) * b)) / (1 - (a * b)) ) ) ) implies f1 = f2 )

assume that
A1: for a, b being Element of [.0,1.] holds
( ( a = b & b = 1 implies f1 . (a,b) = 1 ) & ( ( a <> 1 or b <> 1 ) implies f1 . (a,b) = ((a + b) - ((2 * a) * b)) / (1 - (a * b)) ) ) and
A2: for a, b being Element of [.0,1.] holds
( ( a = b & b = 1 implies f2 . (a,b) = 1 ) & ( ( a <> 1 or b <> 1 ) implies f2 . (a,b) = ((a + b) - ((2 * a) * b)) / (1 - (a * b)) ) ) ; :: thesis: f1 = f2
for a, b being set st a in [.0,1.] & b in [.0,1.] holds
f1 . (a,b) = f2 . (a,b)
proof
let a, b be set ; :: thesis: ( a in [.0,1.] & b in [.0,1.] implies f1 . (a,b) = f2 . (a,b) )
assume ( a in [.0,1.] & b in [.0,1.] ) ; :: thesis: f1 . (a,b) = f2 . (a,b)
then reconsider aa = a, bb = b as Element of [.0,1.] ;
per cases ( ( aa = bb & bb = 1 ) or aa <> 1 or bb <> 1 ) ;
suppose B0: ( aa = bb & bb = 1 ) ; :: thesis: f1 . (a,b) = f2 . (a,b)
then f1 . (aa,bb) = 1 by A1
.= f2 . (aa,bb) by A2, B0 ;
hence f1 . (a,b) = f2 . (a,b) ; :: thesis: verum
end;
suppose B1: ( aa <> 1 or bb <> 1 ) ; :: thesis: f1 . (a,b) = f2 . (a,b)
then f1 . (aa,bb) = ((aa + bb) - ((2 * aa) * bb)) / (1 - (aa * bb)) by A1
.= f2 . (aa,bb) by A2, B1 ;
hence f1 . (a,b) = f2 . (a,b) ; :: thesis: verum
end;
end;
end;
hence f1 = f2 by BINOP_1:def 21; :: thesis: verum