
theorem Th34:
for f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 g be PartFunc of [:REAL,REAL:],REAL, t be Element of REAL
st f is_continuous_on dom f & f = g holds
  ProjPMap1(|.g.|,t) is continuous & ProjPMap2(|.g.|,t) is continuous
proof
    let f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL, t be Element of REAL;
    assume that
A1:  f is_continuous_on dom f and
A2:  f = g;
    set Y = dom ProjPMap1(g,t);

A3: Y = dom |. ProjPMap1(g,t) .| by VALUED_1:def 11;

    ProjPMap1(g,t)|Y is continuous by A1,A2,Th33; then
    (abs ProjPMap1(g,t))|Y is continuous by FCONT_1:21;
    hence ProjPMap1(|.g.|,t) is continuous by A3,Th32;

    set Y2 = dom ProjPMap2(g,t);

A4: Y2 = dom |. ProjPMap2(g,t) .| by VALUED_1:def 11;

    ProjPMap2(g,t)|Y2 is continuous by A1,A2,Th33; then
    (abs ProjPMap2(g,t))|Y2 is continuous by FCONT_1:21;
    hence ProjPMap2(|.g.|,t) is continuous by A4,Th32;
end;
