
theorem Th20:
for z be Element of REAL,
 f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
 Pf2 be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real
  st f is_continuous_on dom f & f = g
   & Pf2 = ProjPMap2(|.R_EAL g.|,z) holds Pf2 is_continuous_on dom Pf2
proof
    let z be Element of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    Pf2 be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
    assume that
A1: f is_continuous_on dom f and
A2: f = g and
A3: Pf2 = ProjPMap2(|.R_EAL g.|,z);

    Pf2 = |. R_EAL(ProjPMap2(g,z)) .| by A3,MESFUN16:31; then
    Pf2 = R_EAL abs(ProjPMap2(g,z)) by MESFUNC6:1; then
    Pf2 = R_EAL ProjPMap2(|.g.|,z) by MESFUN16:32;
    hence Pf2 is_continuous_on dom Pf2 by A1,A2,Th14,MESFUNC5:def 7;
end;
