
theorem Th4:
  for X,Y be RealNormSpace, f be Lipschitzian LinearOperator of X,Y
holds f is_Lipschitzian_on the carrier of X & f is_continuous_on the carrier of
  X & for x be Point of X holds f is_continuous_in x
proof
  let X,Y be RealNormSpace, f be Lipschitzian LinearOperator of X,Y;
  consider K being Real such that
A1: 0 <= K and
A2: for x being VECTOR of X holds ||. f.x .|| <= K * ||. x .|| by
LOPBAN_1:def 8;
A3: now
    let x,y be Point of X;
    assume that
    x in the carrier of X and
    y in the carrier of X;
    f/.x -f/.y =f.x +(-1)* f.y by RLVECT_1:16;
    then f/.x -f/.y = f.x + f.((-1)*y) by LOPBAN_1:def 5;
    then f/.x -f/.y = f.(x+(-1)*y) by VECTSP_1:def 20;
    then
A4: f/.x -f/.y =f.(x+-y) by RLVECT_1:16;
    ||.f/.x -f/.y .||<=K*||. x-y .||+||. x-y .|| by A2,A4,XREAL_1:38;
    hence ||. f/.x -f/.y .|| <= (K+1) * ||. x-y .||;
  end;
  dom f =the carrier of X by FUNCT_2:def 1;
  hence f is_Lipschitzian_on the carrier of X by A1,A3,NFCONT_1:def 9;
  hence
A5: f is_continuous_on the carrier of X by NFCONT_1:45;
  hereby
    let x be Point of X;
    f|(the carrier of X) = f by RELSET_1:19;
    hence f is_continuous_in x by A5,NFCONT_1:def 7;
  end;
end;
