
theorem Th14:
for f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
 p2 be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 z be Element of REAL st f is_continuous_on dom f & f = g
  & p2 = ProjPMap2(|.g.|,z) holds p2 is_continuous_on dom p2
proof
    let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    p2 be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    z be Element of REAL;
    assume that
A1: f is_continuous_on dom f and
A2: f = g and
A3: p2 = ProjPMap2(|.g.|,z);

    reconsider q2 = ProjPMap2(g,z)
      as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
A4: q2 is_continuous_on dom q2 by A1,A2,Th12;

A5: ProjPMap2(|.g.|,z) = |. ProjPMap2(g,z) .| by MESFUN16:32; then
    p2 = ||.q2.|| by A3,MESFUN16:19; then
    p2 is_continuous_on dom q2 by A4,MESFUN16:22;
    hence p2 is_continuous_on dom p2 by A3,A5,VALUED_1:def 11;
end;
