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 Th37:
  for X be RealNormSpace for Y be RealNormSpace for f, g being
Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
 for a be Real holds ( ||.f
  .|| = 0 iff f = 0.R_NormSpace_of_BoundedLinearOperators(X,Y) ) & ||.a*f.|| =
  |.a.| * ||.f.|| & ||.f+g.|| <= ||.f.|| + ||.g.||
proof
  let X be RealNormSpace;
  let Y be RealNormSpace;
  let f, g being Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
  let a be Real;
A1: now
    assume
A2: f = 0.R_NormSpace_of_BoundedLinearOperators(X,Y);
    thus ||.f.|| = 0
    proof
      reconsider g=f as Lipschitzian LinearOperator of X,Y by Def9;
      set z = (the carrier of X) --> 0.Y;
      reconsider z as Function of the carrier of X, the carrier of Y;
      consider r0 be object such that
A3:   r0 in PreNorms(g) by XBOOLE_0:def 1;
      reconsider r0 as Real by A3;
A4:   (for s be Real st s in PreNorms(g) holds s <= 0)
 implies upper_bound
      PreNorms(g) <= 0 by SEQ_4:45;
A5:   PreNorms(g) is non empty bounded_above by Th27;
A6:   z=g by A2,Th31;
A7:   now
        let r be Real;
        assume r in PreNorms(g);
        then consider t be VECTOR of X such that
A8:     r=||.g.t.|| and
        ||.t.|| <= 1;
        ||.g.t.|| = ||.0.Y.|| by A6,FUNCOP_1:7
          .= 0;
        hence 0 <= r & r <=0 by A8;
      end;
      then 0<=r0 by A3;
      then upper_bound PreNorms(g) = 0 by A7,A5,A3,A4,SEQ_4:def 1;
      then BoundedLinearOperatorsNorm(X,Y).f =0 by Th30;
      hence thesis;
    end;
  end;
A9: ||.f+g.|| <= ||.f.|| + ||.g.||
  proof
    reconsider f1=f, g1=g, h1=f+g as Lipschitzian LinearOperator of X,Y
    by Def9;
A10: ( for s be Real st s in PreNorms(h1) holds s <= ||.f.|| + ||.g
    .||) implies upper_bound PreNorms(h1) <= ||.f.|| + ||.g.|| by SEQ_4:45;
A11: now
      let t be VECTOR of X such that
A12:  ||.t.|| <= 1;
      0 <= ||.g.|| by Th33;
      then
A13:  ||.g.||*||.t.|| <= ||.g.||*1 by A12,XREAL_1:64;
      0 <= ||.f.|| by Th33;
      then ||.f.||*||.t.|| <= ||.f.||*1 by A12,XREAL_1:64;
      then
A14:  ||.f.||*||.t.|| + ||.g.||*||.t.|| <= ||.f.||*1 + ||.g.||*1 by A13,
XREAL_1:7;
A15:  ||.f1.t+g1.t.|| <=||.f1.t.||+||.g1.t.|| by NORMSP_1:def 1;
A16:  ||.g1.t.||<= ||.g.||*||.t.|| by Th32;
      ||.f1.t.||<= ||.f.||*||.t.|| by Th32;
      then ||.f1.t.||+||.g1.t.|| <= ||.f.||*||.t.|| + ||.g.||*||.t.|| by A16,
XREAL_1:7;
      then
A17:  ||.f1.t.||+||.g1.t.|| <= ||.f.|| + ||.g.|| by A14,XXREAL_0:2;
      ||.h1.t.||= ||.f1.t+g1.t.|| by Th35;
      hence ||.h1.t.|| <= ||.f.|| + ||.g.|| by A15,A17,XXREAL_0:2;
    end;
A18: now
      let r be Real;
      assume r in PreNorms(h1);
      then ex t be VECTOR of X st r=||.h1.t.|| & ||.t.|| <= 1;
      hence r <= ||.f.|| + ||.g.|| by A11;
    end;
    BoundedLinearOperatorsNorm(X,Y).(f+g) = upper_bound PreNorms(h1) by Th30;
    hence thesis by A18,A10;
  end;
A19: ||.a*f.|| = |.a.| * ||.f.||
  proof
    reconsider f1=f, h1=a*f as Lipschitzian LinearOperator of X,Y by Def9;
A20: (for s be Real st s in PreNorms(h1) holds s <= |.a.|*||.f.||
    ) implies upper_bound PreNorms(h1) <= |.a.|*||.f.|| by SEQ_4:45;
A21: now
A22:  0 <= ||.f.|| by Th33;
      let t be VECTOR of X;
      assume ||.t.|| <= 1;
      then
A23:  ||.f.||*||.t.|| <= ||.f.||*1 by A22,XREAL_1:64;
      ||.f1.t.||<= ||.f.||*||.t.|| by Th32;
      then
A24:  ||.f1.t.|| <= ||.f.|| by A23,XXREAL_0:2;
A25:  ||.a*f1.t.|| =|.a.|*||.f1.t.|| by NORMSP_1:def 1;
A26:  0<= |.a.| by COMPLEX1:46;
      ||.h1.t.||= ||.a*f1.t.|| by Th36;
      hence ||.h1.t.|| <= |.a.|*||.f.|| by A25,A24,A26,XREAL_1:64;
    end;
A27: now
      let r be Real;
      assume r in PreNorms(h1);
      then ex t be VECTOR of X st r=||.h1.t.|| & ||.t.|| <= 1;
      hence r <= |.a.|*||.f.|| by A21;
    end;
A28: now
      per cases;
      case
A29:    a <> 0;
A30:    now
A31:      0 <= ||.a*f.|| by Th33;
          let t be VECTOR of X;
          assume ||.t.|| <= 1;
          then
A32:      ||.a*f.||*||.t.|| <= ||.a*f.||*1 by A31,XREAL_1:64;
          ||.h1.t.||<= ||.a*f.||*||.t.|| by Th32;
          then
A33:      ||.h1.t.|| <= ||.a*f.|| by A32,XXREAL_0:2;
          h1.t=a*f1.t by Th36;
          then
A34:      a"*h1.t =( a"* a)*f1.t by RLVECT_1:def 7
            .=1*f1.t by A29,XCMPLX_0:def 7
            .=f1.t by RLVECT_1:def 8;
A35:      |.a".| =|.1*a".| .=|. 1/a.| by XCMPLX_0:def 9
            .=1/|.a.| by ABSVALUE:7
            .=1*|.a.|" by XCMPLX_0:def 9
            .=|.a.|";
A36:      0<= |.a".| by COMPLEX1:46;
          ||.a"*h1.t.|| =|.a".|*||.h1.t.|| by NORMSP_1:def 1;
          hence ||.f1.t.|| <= |.a.|"*||.a*f.|| by A34,A33,A36,A35,XREAL_1:64;
        end;
A37:    now
          let r be Real;
          assume r in PreNorms(f1);
          then ex t be VECTOR of X st r=||.f1.t.|| & ||.t.|| <= 1;
          hence r <= |.a.|"*||.a*f.|| by A30;
        end;
A38:    ( for s be Real st s in PreNorms(f1) holds s <= |.a.|"*
        ||.a*f.|| ) implies upper_bound PreNorms(f1) <= |.a.|"*||.a*f.||
        by SEQ_4:45;
A39:    0 <= |.a.| by COMPLEX1:46;
        BoundedLinearOperatorsNorm(X,Y).(f) = upper_bound PreNorms(f1) by Th30;
        then ||.f.|| <=|.a.|"*||.a*f.|| by A37,A38;
        then |.a.|*||.f.|| <=|.a.|*(|.a.|"*||.a*f.||) by A39,XREAL_1:64;
        then
A40:    |.a.|*||.f.|| <=(|.a.|*|.a.|")*||.a*f.||;
        |.a.| <>0 by A29,COMPLEX1:47;
        then |.a.|*||.f.|| <=1*||.a*f.|| by A40,XCMPLX_0:def 7;
        hence |.a.|* ||.f.|| <=||.a*f.||;
      end;
      case
A41:    a=0;
        reconsider fz=f as VECTOR of R_VectorSpace_of_BoundedLinearOperators(X
        ,Y);
A42:    a*f =a*fz
          .=0.R_VectorSpace_of_BoundedLinearOperators(X,Y) by A41,RLVECT_1:10
          .=0.R_NormSpace_of_BoundedLinearOperators(X,Y);
        thus |.a.|* ||.f.|| =0 * ||.f.|| by A41,ABSVALUE:2
          .=||.a*f.|| by A42,Th34;
      end;
    end;
    BoundedLinearOperatorsNorm(X,Y).(a*f) = upper_bound PreNorms(h1) by Th30;
    then ||.a*f.|| <= |.a.|*||.f.|| by A27,A20;
    hence thesis by A28,XXREAL_0:1;
  end;
  now
    reconsider g=f as Lipschitzian LinearOperator of X,Y by Def9;
    set z = (the carrier of X) --> 0.Y;
    reconsider z as Function of the carrier of X, the carrier of Y;
    assume
A43: ||.f.|| = 0;
    now
      let t be VECTOR of X;
      ||.g.t.|| <= ||.f.|| *||.t.|| by Th32;
      then ||.g.t.|| = 0 by A43;
      hence g.t =0.Y by NORMSP_0:def 5
        .=z.t by FUNCOP_1:7;
    end;
    then g=z by FUNCT_2:63;
    hence f=0.R_NormSpace_of_BoundedLinearOperators(X,Y) by Th31;
  end;
  hence thesis by A1,A19,A9;
end;
