
theorem Th17:
for x,y be Element of REAL,
 f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL, Pg1 be PartFunc of REAL,REAL
  st f is_continuous_on dom f & f = g
   & Pg1 = ProjPMap1(R_EAL g,[x,y]) holds Pg1 is continuous
proof
    let x,y be Element of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,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,y]);

    Pg1 = R_EAL(ProjPMap1(g,[x,y])) by A3,MESFUN16:31; then
    Pg1 = ProjPMap1(g,[x,y]) by MESFUNC5:def 7;
    hence Pg1 is continuous by A1,A2,Th11;
end;
