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

theorem
  for X, Y be RealNormSpace for g be LinearOperator of X,Y holds g is
  Lipschitzian iff PreNorms(g) is bounded_above
proof
  let X, Y be RealNormSpace;
  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;
          g.t = g.(0*0.X) by A3
            .=0*g.(0.X) by Def5
            .=0.Y by RLVECT_1:10;
          hence ||.g.t.|| <= K*||.t.|| by A4;
        end;
        case
A5:       t <> 0.X;
          reconsider t1= ( ||.t.||")*t as VECTOR of X;
A6:       ||.t.|| <> 0 by A5,NORMSP_0:def 5;
A7:       ||.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.||;
A8:       |. ||.t.||".| = |. 1*||.t.||".| .=|. 1/||.t.||.| by XCMPLX_0:def 9
            .=1/|. ||.t.||.| by ABSVALUE:7
            .=1/||.t.|| by ABSVALUE:def 1
            .=1*||.t.||" by XCMPLX_0:def 9
            .=||.t.||";
          ||.t1.|| =|. ||.t.||".|*||.t.|| by NORMSP_1:def 1
            .=1 by A6,A8,XCMPLX_0:def 7;
          then
A9:      ||.g.t1.|| in {||.g.s.|| where s is VECTOR of X : ||.s.|| <= 1 };
          ||.g.t.||/||.t.|| = ||.g.t.||*||.t.||" by XCMPLX_0:def 9
            .=||. ||.t.||"*g.t.|| by A8,NORMSP_1:def 1
            .=||.g.t1.|| by Def5;
          then ||.g.t.||/||.t.|| <= K by A1,A9,SEQ_4:def 1;
          hence ||.g.t.|| <= K *||.t.|| by A7,XREAL_1:64;
        end;
      end;
      hence ||.g.t.|| <= K*||.t.||;
    end;
    take K;
    0 <= K
    proof
      consider r0 be object such that
A10:  r0 in PreNorms(g) by XBOOLE_0:def 1;
      reconsider r0 as Real by A10;
      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;
      end;
      then 0 <= r0 by A10;
      hence thesis by A1,A10,SEQ_4:def 1;
    end;
    hence g is Lipschitzian by A2;
  end;
  hence thesis by Th27;
end;
