
theorem Th37:
for y be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
  g be PartFunc of [:REAL,REAL:],REAL, Pg2 be PartFunc of REAL,REAL
 st f is_continuous_on dom f & f = g & Pg2 = ProjPMap2(R_EAL g,y) holds
  Pg2 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, Pg2 be PartFunc of REAL,REAL;
    assume that
A1:  f is_continuous_on dom f and
A2:  f = g and
A3:  Pg2 = ProjPMap2(R_EAL g,y);

    Pg2 = R_EAL(ProjPMap2(g,y)) by A3,Th31; then
    Pg2 = ProjPMap2(g,y) by MESFUNC5:def 7;
    hence thesis by A1,A2,Th33;
end;
