let f1, f2 be BinOp of [.0,1.]; :: thesis: ( ( for x, y being Element of [.0,1.] holds
( ( x = y & y = 0 implies f1 . (x,y) = 1 ) & ( ( x > 0 or y > 0 ) implies f1 . (x,y) = y to_power x ) ) ) & ( for x, y being Element of [.0,1.] holds
( ( x = y & y = 0 implies f2 . (x,y) = 1 ) & ( ( x > 0 or y > 0 ) implies f2 . (x,y) = y to_power x ) ) ) implies f1 = f2 )
assume that
A1: for a, b being Element of [.0,1.] holds
( ( a = b & b = 0 implies f1 . (a,b) = 1 ) & ( ( a > 0 or b > 0 ) implies f1 . (a,b) = b to_power a ) ) and
A2: for a, b being Element of [.0,1.] holds
( ( a = b & b = 0 implies f2 . (a,b) = 1 ) & ( ( a > 0 or b > 0 ) implies f2 . (a,b) = b to_power a ) ) ; :: 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.] ;
( aa > 0 or aa = 0 ) by XXREAL_1:1;

per cases then ( ( aa = bb & bb = 0 ) or aa > 0 or bb > 0 ) by XXREAL_1:1;

suppose B0: ( aa = bb & bb = 0 ) ; :: 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 > 0 or bb > 0 ) ; :: thesis: f1 . (a,b) = f2 . (a,b)

then f1 . (aa,bb) = bb to_power aa 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