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

end;

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