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 Th30:
for m,n be non zero Nat, f be PartFunc of REAL m,REAL n,
    f0 be PartFunc of REAL-NS m,REAL-NS n,
    x be Element of REAL m,
    x0 be Element of REAL-NS m st
 x in dom f & x=x0 & f=f0 holds f/.x = f0/.x0
proof
   let m,n be non zero Nat,
       f be PartFunc of REAL m,REAL n,
       f0 be PartFunc of REAL-NS m,REAL-NS n,
       x be Element of REAL m, x0 be Element of REAL-NS m;
   assume A1: x in dom f & x=x0 & f=f0;
   then f/.x = f0.x0 by PARTFUN1:def 6;
   hence f/.x = f0/.x0 by A1,PARTFUN1:def 6;
end;
