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

theorem Th43:
  X c= dom f & f|X is constant implies f is_Lipschitzian_on X
proof
  assume that
A1: X c= dom f and
A2: f|X is constant;
  now
    let x1,x2;
    assume that
A3: x1 in X and
A4: x2 in X;
A5: x1 in X /\ dom f & x2 in X /\ dom f by A1,A3,A4,XBOOLE_0:def 4;
    f/.x1= f.x1 by A1,A3,PARTFUN1:def 6
      .= f.x2 by A2,A5,PARTFUN2:58
      .= f/.x2 by A1,A4,PARTFUN1:def 6;
    then ||. f/.x1-f/.x2.|| = ||. 0.T .|| by RLVECT_1:15
      .= 0 by NORMSP_1:1;
    hence ||. f/.x1-f/.x2.|| <= 1*||. x1-x2 .|| by NORMSP_1:4;
  end;
  hence thesis by A1;
end;
