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 Th32:
  for X, Y be RealNormSpace for f being Point of
R_NormSpace_of_BoundedLinearOperators(X,Y)
for g be Lipschitzian LinearOperator of X
  ,Y st g=f holds for t be VECTOR of X holds ||.g.t.|| <= ||.f.|| * ||.t.||
proof
  let X, Y be RealNormSpace;
  let f being Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
  let g be Lipschitzian LinearOperator of X,Y such that
A1: g=f;
A2: PreNorms(g) is non empty bounded_above by Th27;
  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.|| <= ||.f.||*||.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:     |. ||.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.||";
A8:     ||.g.t.||/||.t.|| = ||.g.t.||*||.t.||" by XCMPLX_0:def 9
          .=||. ||.t.||"*g.t.|| by A7,NORMSP_1:def 1
          .=||.g.t1.|| by Def5;
        ||.t1.|| =|. ||.t.||".|*||.t.|| by NORMSP_1:def 1
          .=1 by A6,A7,XCMPLX_0:def 7;
        then ||.g.t.||/||.t.|| in {||.g.s.|| where s is VECTOR of X : ||.s.||
        <= 1 } by A8;
        then ||.g.t.||/||.t.|| <= upper_bound PreNorms(g) by A2,SEQ_4:def 1;
        then ||.g.t.||/||.t.|| <= BoundedLinearOperatorsNorm(X,Y).g by Th30;
        then
A9:    ||.g.t.||/||.t.|| <= ||.f.|| by A1;
        ||.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.|| <= ||.f.||*||.t.|| by A9,XREAL_1:64;
      end;
    end;
    hence ||.g.t.|| <= ||.f.||*||.t.||;
  end;
  hence thesis;
end;
