
theorem Th9:
for f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL st f = g holds ||.f.|| = |.g.|
proof
    let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
        g be PartFunc of [:[:REAL,REAL:],REAL:],REAL;
    assume
A1:  f = g;

A2: dom ||.f.|| = dom f by NORMSP_0:def 3; then
A3: dom ||.f.|| = dom |.g.| by A1,VALUED_1:def 11;
    for x be object st x in dom ||.f.|| holds (||.f.||).x = (|.g.|).x
    proof
     let x be object;
     assume
A4:   x in dom ||.f.||; then
A5:  (||.f.||).x = ||. f/.x .|| by NORMSP_0:def 3;

     f/.x = f.x by A4,A2,PARTFUN1:def 6; then
     ||. f/.x .|| = |. g.x .| by A1,EUCLID:def 2;
     hence (||.f.||).x = (|.g.|).x by A3,A4,A5,VALUED_1:def 11;
    end;
    hence ||.f.|| = |.g.| by A3,FUNCT_1:2;
end;
