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) `2 ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = (x `2) `2 ) implies f = g )
assume that
A11: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = (x `2) `2 and
A12: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = (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 `2) `2 by A11
.= g . x by A12 ; :: thesis: verum

end;

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