let f, g be RealMap of [:(TOP-REAL 2),(TOP-REAL 2):]; :: thesis: ( ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `1) `2) - ((x `2) `2) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `1) `2) - ((x `2) `2) ) implies f = g )
assume that
A5: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `1) `2) - ((x `2) `2) and
A6: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `1) `2) - ((x `2) `2) ; :: thesis: f = g

now :: thesis: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = g . x

let x be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; :: thesis: f . x = g . x
thus f . x = ((x `1) `2) - ((x `2) `2) by A5
.= g . x by A6 ; :: thesis: verum

end;

hence f = g by FUNCT_2:63; :: thesis: verum