
theorem Th21:
  for X be non empty set for Y be RealNormSpace for f, g being
Point of R_NormSpace_of_BoundedFunctions(X,Y)
 for a be Real holds ( ||.f.|| = 0
iff f = 0.R_NormSpace_of_BoundedFunctions(X,Y) ) & ||.a*f.|| = |.a.| * ||.f.||
  & ||.f+g.|| <= ||.f.|| + ||.g.||
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let f, g being Point of R_NormSpace_of_BoundedFunctions(X,Y);
  let a be Real;
A1: now
    assume
A2: f = 0.R_NormSpace_of_BoundedFunctions(X,Y);
    thus ||.f.|| = 0
    proof
      reconsider g=f as bounded Function of X,the carrier of Y by Def5;
      set z = X --> 0.Y;
      reconsider z as Function 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 Th11;
A6:   z=g by A2,Th15;
A7:   now
        let r be Real;
        assume r in PreNorms(g);
        then consider t be Element of X such that
A8:     r=||.g.t.||;
        ||.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;
      hence thesis by Th14;
    end;
  end;
A9: ||.f+g.|| <= ||.f.|| + ||.g.||
  proof
    reconsider f1=f, g1=g, h1=f+g as bounded Function of X,the carrier of Y by
Def5;
A10: now
      let t be Element of X;
      ||.f1.t.||<= ||.f.|| & ||.g1.t.||<= ||.g.|| by Th16;
      then
A11:  ||.f1.t.||+||.g1.t.|| <= ||.f.|| + ||.g.|| by XREAL_1:7;
      ||.h1.t.||= ||.f1.t+g1.t.|| & ||.f1.t+g1.t.|| <=||.f1.t.||+||.g1.t
      .|| by Th19,NORMSP_1:def 1;
      hence ||.h1.t.|| <= ||.f.|| + ||.g.|| by A11,XXREAL_0:2;
    end;
A12: now
      let r be Real;
      assume r in PreNorms(h1);
      then ex t be Element of X st r=||.h1.t.||;
      hence r <= ||.f.|| + ||.g.|| by A10;
    end;
    (for s be Real st s in PreNorms(h1) holds s <= ||.f.|| + ||.g
    .||) implies upper_bound PreNorms(h1) <= ||.f.|| + ||.g.|| by SEQ_4:45;
    hence thesis by A12,Th14;
  end;
A13: ||.a*f.|| = |.a.| * ||.f.||
  proof
    reconsider f1=f, h1=a*f as bounded Function of X, the carrier of Y by Def5;
A14: (for s be Real st s in PreNorms(h1) holds s <= |.a.|*||.f.||
    ) implies upper_bound PreNorms(h1) <= |.a.|*||.f.|| by SEQ_4:45;
A15: now
      let t be Element of X;
A16:  0<= |.a.| by COMPLEX1:46;
      ||.h1.t.||= ||.a*f1.t.|| & ||.a*f1.t.|| =|.a.|*||.f1.t.|| by Th20,
NORMSP_1:def 1;
      hence ||.h1.t.|| <= |.a.|*||.f.|| by A16,Th16,XREAL_1:64;
    end;
A17: now
      let r be Real;
      assume r in PreNorms(h1);
      then ex t be Element of X st r=||.h1.t.||;
      hence r <= |.a.|*||.f.|| by A15;
    end;
A18: now
      per cases;
      case
A19:    a <> 0;
A20:    now
          let t be Element of X;
A21:      |.a".| =|.1*a".| .=|. 1/a.| by XCMPLX_0:def 9
            .=1/|.a.| by ABSVALUE:7
            .=1*|.a.|" by XCMPLX_0:def 9
            .=|.a.|";
          h1.t=a*f1.t by Th20;
          then
A22:      a"*h1.t =( a"* a)*f1.t by RLVECT_1:def 7
            .=1*f1.t by A19,XCMPLX_0:def 7
            .=f1.t by RLVECT_1:def 8;
          ||.a"*h1.t.|| =|.a".|*||.h1.t.|| & 0<= |.a".| by COMPLEX1:46
,NORMSP_1:def 1;
          hence ||.f1.t.|| <= |.a.|"*||.a*f.|| by A22,A21,Th16,XREAL_1:64;
        end;
A23:    now
          let r be Real;
          assume r in PreNorms(f1);
          then ex t be Element of X st r=||.f1.t.||;
          hence r <= |.a.|"*||.a*f.|| by A20;
        end;
A24:    ( 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;
        BoundedFunctionsNorm(X,Y).(f) = upper_bound PreNorms(f1) & 0 <= |.a.|
         by Th14,COMPLEX1:46;
        then |.a.|*||.f.|| <=|.a.|*(|.a.|"*||.a*f.||) by A23,A24,XREAL_1:64;
        then
A25:    |.a.|*||.f.|| <=(|.a.|*|.a.|")*||.a*f.||;
        |.a.| <>0 by A19,COMPLEX1:47;
        then |.a.|*||.f.|| <=1*||.a*f.|| by A25,XCMPLX_0:def 7;
        hence |.a.|* ||.f.|| <=||.a*f.||;
      end;
      case
A26:    a=0;
        reconsider fz=f as VECTOR of R_VectorSpace_of_BoundedFunctions(X,Y);
A27:    a*f =a*fz .=0.R_VectorSpace_of_BoundedFunctions(X,Y) by A26,RLVECT_1:10
          .=0.R_NormSpace_of_BoundedFunctions(X,Y);
        thus |.a.|* ||.f.|| =0 * ||.f.|| by A26,ABSVALUE:2
          .=||.a*f.|| by A27,Th18;
      end;
    end;
    BoundedFunctionsNorm(X,Y).(a*f) = upper_bound PreNorms(h1) by Th14;
    hence thesis by A17,A14,A18,XXREAL_0:1;
  end;
  now
    reconsider g=f as bounded Function of X,the carrier of Y by Def5;
    set z = X --> 0.Y;
    reconsider z as Function of X, the carrier of Y;
    assume
A28: ||.f.|| = 0;
    now
      let t be Element of X;
      ||.g.t.|| <= ||.f.|| by Th16;
      then ||.g.t.|| = 0 by A28;
      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_BoundedFunctions(X,Y) by Th15;
  end;
  hence thesis by A1,A13,A9;
end;
