let V1, V2 be Function of NAT,ExtREAL; :: thesis: ( ( for n being Nat holds V1 . n = M . (FSets . n) ) & ( for n being Nat holds V2 . n = M . (FSets . n) ) implies V1 = V2 )
assume that
A2: for n being Nat holds V1 . n = M . (FSets . n) and
A3: for n being Nat holds V2 . n = M . (FSets . n) ; :: thesis: V1 = V2

now :: thesis: for x being set st x in NAT holds
V1 . x = V2 . x

let x be set ; :: thesis: ( x in NAT implies V1 . x = V2 . x )
assume x in NAT ; :: thesis: V1 . x = V2 . x
then reconsider n = x as Nat ;
thus V1 . x = M . (FSets . n) by A2
.= V2 . x by A3 ; :: thesis: verum

end;

hence V1 = V2 by FUNCT_2:12; :: thesis: verum