
theorem Th39:
for y be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 g be PartFunc of [:REAL,REAL:],REAL, p2 be PartFunc of REAL,REAL
  st f is_continuous_on dom f & f = g & p2 = ProjPMap2(|.R_EAL g.|,y)
  holds p2 is continuous
proof
    let y be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL, p2 be PartFunc of REAL,REAL;
    assume that
A1: f is_continuous_on dom f and
A2: f = g and
A3: p2 = ProjPMap2(|.R_EAL g.|,y);

    p2 = |. R_EAL(ProjPMap2(g,y)) .| by A3,Th31; then
    p2 = R_EAL abs(ProjPMap2(g,y)) by MESFUNC6:1; then
    p2 = R_EAL ProjPMap2(|.g.|,y) by Th32; then
    p2 = ProjPMap2(|.g.|,y) by MESFUNC5:def 7;
    hence thesis by A1,A2,Th34;
end;
