reserve V for non empty RealLinearSpace;
reserve S for Real_Sequence;
reserve k,n,m,m1 for Nat;
reserve g,h,r,x for Real;

theorem
  for X be RealNormSpace, g be linear-Functional of X
    holds g is Lipschitzian iff PreNorms g is bounded_above
proof
  let X be RealNormSpace;
  let g be linear-Functional of X;
  now
    reconsider K=upper_bound PreNorms g as Real;
    assume
A1: PreNorms g is bounded_above;
A2: now
     let t be VECTOR of X;
     per cases;
     suppose
A3:   t = 0.X; then
A4:   ||.t.|| = 0;
      g.t = g.(0*0.X) by A3
         .=0*g.(0.X) by HAHNBAN:def 3
         .=0;
      hence |.g.t.| <= K*||.t.|| by A4,COMPLEX1:44;
     end;
     suppose
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: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,COMPLEX1:65
            .= |.g.t1.| by HAHNBAN:def 3;
      then |.g.t.|/||.t.|| <= K by A1,A9,SEQ_4:def 1;
      hence |.g.t.| <= K *||.t.|| by A7,XREAL_1:64;
     end;
    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 by COMPLEX1:46;
     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;
end;
