reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem Th6:
x0 in X & f is_continuous_in x0 implies f|X is_continuous_in x0
proof
   assume that
A1: x0 in X and
A2: f is_continuous_in x0;
A3:x0 in dom f by A2;
A4:dom(f|X) = X /\ dom f by RELAT_1:61;
   hence
A5: x0 in dom (f|X) by A1,A3,XBOOLE_0:def 4;
  let s1 such that
A6: rng s1 c= dom(f|X) and
A7: s1 is convergent & lim s1 = x0;
A8:rng s1 c= dom f by A6,A4,XBOOLE_1:18;
A9:(f|X)/*s1 = f/*s1 by A6,FUNCT_2:117;
   hence (f|X)/*s1 is convergent by A2,A7,A8;
    x0 in REAL by XREAL_0:def 1;
   hence (f|X)/.x0 = f/.x0 by A5,PARTFUN2:15
             .= lim ((f|X)/*s1) by A2,A7,A8,A9;
end;
