reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LMCONT1:
  for E,F be RealNormSpace,
        L be Lipschitzian LinearOperator of E,F
  holds
    L is_Lipschitzian_on the carrier of E &
    L is_uniformly_continuous_on the carrier of E
  proof
    let E,F be RealNormSpace,
          L be Lipschitzian LinearOperator of E,F;
    consider K being Real such that
    A2: 0 <= K &
          for x being VECTOR of E holds ||. L.x.|| <= K * ||.x.||
            by LOPBAN_1:def 8;
    set r = K+1;
    A3: K + 0 < r by XREAL_1:8;
    set E0 = the carrier of E;
    for x1, x2 being Point of E st x1 in E0 & x2 in E0 holds
    ||.L/. x1 - L/. x2.|| <= r * ||. x1 -x2.||
    proof
      let x1, x2 be Point of E;
      L/.x1 - L/.x2 = L.x1+(-1)*(L.x2) by RLVECT_1:16
       .= L.x1+L. ((-1)*x2) by LOPBAN_1:def 5
       .= L.(x1+(-1)*x2) by VECTSP_1:def 20
       .= L.(x1-x2) by RLVECT_1:16; then
      A8: ||.L/. x1 - L/. x2.|| <= K * ||.x1-x2.|| by A2;
      K * ||.x1-x2.|| <= r * ||.x1-x2.|| by A3,XREAL_1:64;
      hence thesis by A8,XXREAL_0:2;
    end;
    hence L is_Lipschitzian_on E0 by A2,FUNCT_2:def 1;
    hence L is_uniformly_continuous_on E0 by NFCONT_2:9;
  end;
