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 ) `1 ) - ((x `2 ) `1 ) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `1 ) `1 ) - ((x `2 ) `1 ) ) implies f = g )
assume that
A2: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `1 ) `1 ) - ((x `2 ) `1 ) and
A3: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `1 ) `1 ) - ((x `2 ) `1 ) ; :: thesis: f = g

now

let x be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; :: thesis: f . x = g . x
thus f . x = ((x `1 ) `1 ) - ((x `2 ) `1 ) by A2
.= g . x by A3 ; :: thesis: verum

end;

hence f = g by FUNCT_2:113; :: thesis: verum