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
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 Element of REAL m,
    y be Point of REAL-NS m
 st f=g & x=y holds
f is_continuous_in x iff g is_continuous_in y;
