let S1, S2 be Function of [: the carrier of F,(NonZero F):], the carrier of F; :: thesis: ( ( for x being Element of F
for y being Element of NonZero F holds S1 . (x,y) = (omf F) . (x,((revf F) . y)) ) & ( for x being Element of F
for y being Element of NonZero F holds S2 . (x,y) = (omf F) . (x,((revf F) . y)) ) implies S1 = S2 )
assume that
A2: for x being Element of F
for y being Element of NonZero F holds S1 . (x,y) = (omf F) . (x,((revf F) . y)) and
A3: for x being Element of F
for y being Element of NonZero F holds S2 . (x,y) = (omf F) . (x,((revf F) . y)) ; :: thesis: S1 = S2

now :: thesis: for p being object st p in [: the carrier of F,(NonZero F):] holds
S1 . p = S2 . p

let p be object ; :: thesis: ( p in [: the carrier of F,(NonZero F):] implies S1 . p = S2 . p )
assume p in [: the carrier of F,(NonZero F):] ; :: thesis: S1 . p = S2 . p
then consider x, y being object such that
A4: ( x in the carrier of F & y in NonZero F ) and
A5: p = [x,y] by ZFMISC_1:def 2;
S1 . (x,y) = (omf F) . (x,((revf F) . y)) by A2, A4
.= S2 . (x,y) by A3, A4 ;
hence S1 . p = S2 . p by A5; :: thesis: verum

end;

hence S1 = S2 by FUNCT_2:12; :: thesis: verum