reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th37:
for m,n be non zero Nat, f be PartFunc of REAL m,REAL n,
    g be PartFunc of REAL-NS m,REAL-NS n,
    X be set st f=g holds
 f is_continuous_on X iff g is_continuous_on X
proof
   let m,n be non zero Nat,
       f be PartFunc of REAL m,REAL n,
       g be PartFunc of REAL-NS m,REAL-NS n,
       X be set;
   assume A1: f=g;
   hereby assume A2: f is_continuous_on X;
then A3: X c= dom f & for x0 be Element of REAL m st x0 in X holds
      f|X is_continuous_in x0;
     now let y0 be Point of REAL-NS m;
     assume A4: y0 in X;
     reconsider x0=y0 as Element of REAL m by REAL_NS1:def 4;
      f|X is_continuous_in x0 by A2,A4;
     hence g|X is_continuous_in y0 by A1;
    end;
    hence g is_continuous_on X by A3,A1,NFCONT_1:def 7;
   end;
   assume A5: g is_continuous_on X;
then A6:X c= dom g & for y0 be Point of REAL-NS m st y0 in X holds
     g|X is_continuous_in y0 by NFCONT_1:def 7;
    now let x0 be Element of REAL m;
    assume A7: x0 in X;
    reconsider y0=x0 as Point of REAL-NS m by REAL_NS1:def 4;
     g|X is_continuous_in y0 by A5,A7,NFCONT_1:def 7;
    hence f|X is_continuous_in x0 by A1;
   end;
   hence f is_continuous_on X by A6,A1;
end;
