reserve X for non empty set;
reserve Y for ComplexLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Complex;
reserve u,v,w for VECTOR of CLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem
  for X,Y be ComplexNormSpace, g be LinearOperator of X,Y holds g is
  Lipschitzian iff PreNorms(g) is bounded_above
proof
  let X,Y be ComplexNormSpace;
  let g be LinearOperator of X,Y;
  now
    reconsider K=upper_bound PreNorms(g) as Real;
    assume
A1: PreNorms(g) is bounded_above;
A2: now
      let t be VECTOR of X;
      now
        per cases;
        case
A3:       t = 0.X;
          then
A4:       ||.t.|| = 0 by NORMSP_0:def 6;
          g.t = g.(0c*0.X) by A3,CLVECT_1:1
            .=0c*g.(0.X) by Def3
            .=0.Y by CLVECT_1:1;
          hence ||.g.t.|| <= K*||.t.|| by A4,NORMSP_0:def 6;
        end;
        case
A5:       t <> 0.X;
          reconsider t0 = ||.t.||"+0*<i> as Element of COMPLEX by
XCMPLX_0:def 2;
          reconsider t1= t0 * t as VECTOR of X;
A6:       ||.t.|| <> 0 by A5,NORMSP_0:def 5;
          then
A7:       ||.t.|| > 0 by CLVECT_1:105;
A8:       |. ||.t.||"+0*<i> .| =|. 1*||.t.||" .|
            .=|. 1/||.t.||.| by XCMPLX_0:def 9
            .=1/|. ||.t.||.| by ABSVALUE:7
            .=1/||.t.|| by A7,ABSVALUE:def 1
            .=1*||.t.||" by XCMPLX_0:def 9
            .=||.t.||";
          then
A9:       ||.g.t.||/||.t.|| =||.g.t.|| * |. t0 .| by XCMPLX_0:def 9
            .=||. t0*g.t .|| by CLVECT_1:def 13
            .=||.g.t1.|| by Def3;
          ||.t1.|| = |.t0.| * ||.t.|| by CLVECT_1:def 13
            .=1 by A6,A8,XCMPLX_0:def 7;
          then ||.g.t.||/||.t.|| in PreNorms(g) by A9;
          then
A10:      ||.g.t.||/||.t.|| <= K by A1,SEQ_4:def 1;
          ||.g.t.||/||.t.||*||.t.|| = ||.g.t.||*||.t.||"*||.t.|| by
XCMPLX_0:def 9
            .=||.g.t.||*(||.t.||"*||.t.||)
            .=||.g.t.||*1 by A6,XCMPLX_0:def 7
            .=||.g.t.||;
          hence ||.g.t.|| <= K *||.t.|| by A7,A10,XREAL_1:64;
        end;
      end;
      hence ||.g.t.|| <= K*||.t.||;
    end;
    take K;
    0 <= K
    proof
      consider r0 be object such that
A11:  r0 in PreNorms(g) by XBOOLE_0:def 1;
      reconsider r0 as Real by A11;
      now
        let r be Real;
        assume r in PreNorms(g);
        then ex t be VECTOR of X st r=||.g.t.|| & ||.t.|| <= 1;
        hence 0 <= r by CLVECT_1:105;
      end;
      then 0 <= r0 by A11;
      hence thesis by A1,A11,SEQ_4:def 1;
    end;
    hence g is Lipschitzian by A2;
  end;
  hence thesis by Th26;
end;
