let V1, V2 be sequence of 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

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 object st x in NAT holds

V1 . x = V2 . x

hence
V1 = V2
by FUNCT_2:12; :: thesis: verumV1 . x = V2 . x

let x be object ; :: 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;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