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 ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = (x `2) `1 ) implies f = g )
assume that
A8: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = (x `2) `1 and
A9: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = (x `2) `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 by A8
.= g . x by A9 ; :: thesis: verum

end;

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