
theorem Th38:
for x be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 g be PartFunc of [:REAL,REAL:],REAL, Pg1 be PartFunc of REAL,REAL
  st f is_continuous_on dom f & f = g & Pg1 = ProjPMap1(|.R_EAL g.|,x)
  holds Pg1 is continuous
proof
    let x be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL, Pg1 be PartFunc of REAL,REAL;
    assume that
A1: f is_continuous_on dom f and
A2: f = g and
A3: Pg1 = ProjPMap1(|.R_EAL g.|,x);

    Pg1 = |. R_EAL(ProjPMap1(g,x)) .| by A3,Th31; then
    Pg1 = R_EAL abs(ProjPMap1(g,x)) by MESFUNC6:1; then
    Pg1 = R_EAL ProjPMap1(|.g.|,x) by Th32; then
    Pg1 = ProjPMap1(|.g.|,x) by MESFUNC5:def 7;
    hence thesis by A1,A2,Th34;
end;
