reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th44:
for f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1
 st <>*f = g holds Z c= dom f & f is_continuous_on Z iff g is_continuous_on Z
proof
   let f be PartFunc of REAL m,REAL, g be PartFunc of REAL m,REAL 1;
   assume A1: <>*f=g; then
A2:<>*(f|Z) =g|Z by Th5;
   hence Z c= dom f & f is_continuous_on Z implies g is_continuous_on Z
   by A1,Th3,Th37;
   assume A3: g is_continuous_on Z;
   hence Z c= dom f by Th3,A1;
   let x0 be Element of REAL m;
   assume x0 in Z;
   hence f|Z is_continuous_in x0 by A3,A2,Th37;
end;
