reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th41:
  (for x0 st x0 in X holds f.x0 = r*x0+p) implies f|X is continuous
proof
  assume
A1: for x0 st x0 in X holds f.x0 = r*x0+p;
A2: now
    let x1,x2;
    assume that
A3: x1 in dom(f|X) and
A4: x2 in dom(f|X);
    x2 in X by A4;
    then
A5: f.x2 = r*x2+p by A1;
A6: 0<=|.x1-x2.| by COMPLEX1:46;
    x1 in X by A3;
    then f.x1 = r*x1+p by A1;
    then |.f.x1-f.x2.| = |.r*(x1-x2).| by A5
      .= |.r.|*|.x1-x2.| by COMPLEX1:65;
    then |.f.x1-f.x2.| + (0 qua Nat) <= |.r.|*|.x1-x2.| + 1*|.x1-x2.| by A6
,XREAL_1:7;
    hence |.f.x1-f.x2.| <= (|.r.|+1)*|.x1-x2.|;
  end;
  (0 qua Nat)+(0 qua Nat)<|.r.|+1 by COMPLEX1:46,XREAL_1:8;
  then f|X is Lipschitzian by A2,Th32;
  hence thesis;
end;
