reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;
reserve F,G,H for VECTOR of R_Algebra_of_BoundedFunctions X;
reserve f,g,h for Function of X,REAL;
reserve F,G,H for Point of R_Normed_Algebra_of_BoundedFunctions X;

theorem Th32:
  ( ||.F.|| = 0 iff F = 0.R_Normed_Algebra_of_BoundedFunctions X )
  & ||.a*F.|| = |.a.| * ||.F.|| & ||.F+G.|| <= ||.F.|| + ||.G.||
proof
A1: now
    set z = X --> In(0,REAL);
    reconsider z as Function of X,REAL;
    F in BoundedFunctions X;
    then consider g be Function of X,REAL such that
A2: F=g and
A3: g|X is bounded;
A4: PreNorms g is non empty bounded_above by A3,Th17;
    consider r0 be object such that
A5: r0 in PreNorms g by XBOOLE_0:def 1;
    reconsider r0 as Real by A5;
A6: (for s be Real st s in PreNorms g holds s <= 0) implies upper_bound
    PreNorms g <= 0 by SEQ_4:45;
    assume F = 0.R_Normed_Algebra_of_BoundedFunctions X;
    then
A7: z=g by A2,Th25;
A8: now
      let r be Real;
      assume r in PreNorms g;
      then consider t be Element of X such that
A9:   r=|.g.t.|;
      |.g.t.| = |.0 .| by A7
        .= 0 by ABSVALUE:2;
      hence 0 <= r & r <=0 by A9;
    end;
    then 0<=r0 by A5;
    then upper_bound PreNorms g = 0 by A8,A4,A5,A6,SEQ_4:def 1;
    hence ||.F.|| = 0 by A2,A3,Th20;
  end;
A10: ||.F+G.|| <= ||.F.|| + ||.G.||
  proof
    F+G in BoundedFunctions X;
    then consider h1 be Function of X,REAL such that
A11: h1=F+G and
A12: h1|X is bounded;
    G in BoundedFunctions X;
    then consider g1 be Function of X,REAL such that
A13: g1=G and
A14: g1|X is bounded;
    F in BoundedFunctions X;
    then consider f1 be Function of X,REAL such that
A15: f1=F and
A16: f1|X is bounded;
A17: now
      let t be Element of X;
      |.f1.t.| <= ||.F.|| & |.g1.t.|<= ||.G.|| by A15,A16,A13,A14,Th26;
      then
A18:  |.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 A15,A13,A11,Th29,COMPLEX1:56;
      hence |.h1.t.| <= ||.F.|| + ||.G.|| by A18,XXREAL_0:2;
    end;
A19: 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 A17;
    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 A11,A12,A19,Th20;
  end;
A20: ||.a*F.|| = |.a.| * ||.F.||
  proof
    F in BoundedFunctions X;
    then consider f1 be Function of X,REAL such that
A21: f1=F and
A22: f1|X is bounded;
    a*F in BoundedFunctions X;
    then consider h1 be Function of X,REAL such that
A23: h1=a*F and
A24: h1|X is bounded;
A25: now
      let t be Element of X;
      |.h1.t.| = |.a*f1.t.| by A21,A23,Th30;
      then
A26:  |.h1.t.| = |.a.| * |.f1.t.| by COMPLEX1:65;
      0<= |.a.| by COMPLEX1:46;
      hence |.h1.t.| <= |.a.| *||.F.|| by A21,A22,A26,Th26,XREAL_1:64;
    end;
A27: 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 A25;
    end;
    (for s be Real st s in PreNorms h1 holds s <= |.a.| *||.F.|| )
    implies upper_bound PreNorms h1 <= |.a.| *||.F.|| by SEQ_4:45;
    then
A28: ||.a*F.|| <= |.a.| *||.F.|| by A23,A24,A27,Th20;
    per cases;
    suppose
A29:  a <> 0;
A30:  now
        let t be Element of X;
        |.a".| =|.1/a.|;
        then
A31:    |.a".| =1/|.a.| by ABSVALUE:7;
        h1.t=a*f1.t by A21,A23,Th30;
        then a"*h1.t = (a"* a)*f1.t;
        then
A32:    a"*h1.t =1*f1.t by A29,XCMPLX_0:def 7;
        |.a"*h1.t.| = |.a".|*|.h1.t.| & 0 <= |.a".| by COMPLEX1:46,65;
        hence |.f1.t.| <= |.a.|"*||.a*F.|| by A23,A24,A32,A31,Th26,XREAL_1:64;
      end;
A33:  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 A30;
      end;
A34:  0 <= |.a.| by COMPLEX1:46;
      (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;
      then ||.F.|| <=|.a.|"*||.a*F.|| by A21,A22,A33,Th20;
      then |.a.|*||.F.|| <=|.a.|*(|.a.|"*||.a*F.||) by A34,XREAL_1:64;
      then
A35:  |.a.|*||.F.|| <=(|.a.|*|.a.|")*||.a*F.||;
      |.a.| <>0 by A29,COMPLEX1:47;
      then |.a.|*||.F.|| <=1*||.a*F.|| by A35,XCMPLX_0:def 7;
      hence thesis by A28,XXREAL_0:1;
    end;
    suppose
A36:  a=0;
      reconsider fz=F as VECTOR of R_Algebra_of_BoundedFunctions X;
      a*fz =(a+a)*fz by A36
        .=a*fz + a* fz by RLVECT_1:def 6;
      then 0.R_Algebra_of_BoundedFunctions X = -(a*fz)+(a*fz + a* fz) by
VECTSP_1:16;
      then 0.R_Algebra_of_BoundedFunctions X = -(a*fz)+a*fz + a* fz by
RLVECT_1:def 3;
      then
      0.R_Algebra_of_BoundedFunctions X = 0.R_Algebra_of_BoundedFunctions
      X + a * fz by VECTSP_1:16;
      then
A37:  a*F =0.R_Normed_Algebra_of_BoundedFunctions X;
      |.a.|* ||.F.|| =0 * ||.F.|| by A36,ABSVALUE:2;
      hence thesis by A37,Th28;
    end;
  end;
  now
    set z = X --> In(0,REAL);
    reconsider z as Function of X,REAL;
    F in BoundedFunctions X;
    then consider g be Function of X,REAL such that
A38: F=g and
A39: g|X is bounded;
    assume
A40: ||.F.|| = 0;
    now
      let t be Element of X;
      |.g.t.| <= ||.F.|| by A38,A39,Th26;
      then |.g.t.| = 0 by A40,COMPLEX1:46;
      hence g.t =0 by ABSVALUE:2
        .=z.t;
    end;
    then g=z by FUNCT_2:63;
    hence F=0.R_Normed_Algebra_of_BoundedFunctions X by A38,Th25;
  end;
  hence thesis by A1,A20,A10;
end;
