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
for r, p be Point of S st (for x0 st x0 in X holds f/.x0 = x0*r+p)
 holds f|X is continuous
proof
   let r, p be Point of S;
   assume A1: for x0 st x0 in X holds f/.x0 = x0*r+p;
A2:now let x1,x2;
    assume A3: x1 in dom(f|X) & x2 in dom(f|X); then
    x2 in X; then
A4: f/.x2 = x2*r+p by A1;
A5: 0<=|.x1-x2.| by COMPLEX1:46;
    x1 in X by A3; then
    f/.x1 = x1*r+p by A1; then
    ||. f/.x1-f/.x2.|| = ||. x1*r+(p-(p+x2*r)).|| by A4,RLVECT_1:def 3
      .= ||. x1*r+(p-p-x2*r).|| by RLVECT_1:27
      .= ||. x1*r+(0.S-x2*r).|| by RLVECT_1:15
      .= ||. x1*r-x2*r.|| by RLVECT_1:14
      .= ||. (x1-x2)*r.|| by RLVECT_1:35
      .= |.x1-x2.|* ||. r .|| by NORMSP_1:def 1; then
    ||.f/.x1-f/.x2.|| + (0 qua Nat) <= ||.r.||*|.x1-x2.| + 1*|.x1-x2.|
        by A5,XREAL_1:7;
    hence ||.f/.x1-f/.x2.|| <= (||.r.||+1)*|.x1-x2.|;
   end;
   (0 qua Nat)+(0 qua Nat) < ||.r.|| +1 by NORMSP_1:4,XREAL_1:8; then
   f|X is Lipschitzian by A2,Th26;
   hence thesis;
end;
