
theorem Th36:
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 = ProjPMap1(g,x) by MESFUNC5:def 7;
    hence thesis by A1,A2,Th33;
end;
