
theorem Th13:
for f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
 x,y be Element of REAL st f is_continuous_on dom f & f = g holds
  ProjPMap1(|.g.|,[x,y]) is continuous
proof
    let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    x,y be Element of REAL;
    assume that
A1: f is_continuous_on dom f and
A2: f = g;

    set Y = dom ProjPMap1(g,[x,y]);
A3: Y = dom |. ProjPMap1(g,[x,y]) .| by VALUED_1:def 11;

    ProjPMap1(g,[x,y])|Y is continuous by A1,A2,Th11; then
    (abs ProjPMap1(g,[x,y]))|Y is continuous by FCONT_1:21;
    hence ProjPMap1(|.g.|,[x,y]) is continuous by A3,MESFUN16:32;
end;
