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 Th34:
  f=F & g=G & h=H implies (H = F-G iff for x be Element of X holds
  h.x = f.x - g.x )
proof
  assume
A1: f=F & g=G & h=H;
A2: now
    assume
A3: for x be Element of X holds h.x = f.x - g.x;
    now
      let x be Element of X;
      h.x = f.x - g.x by A3;
      hence h.x + g.x= f.x;
    end;
    then F=H+G by A1,Th29;
    then F-G=H+(G-G) by RLVECT_1:def 3;
    then F-G=H+0.R_Normed_Algebra_of_BoundedFunctions X by RLVECT_1:15;
    hence F-G=H;
  end;
  now
    assume H=F-G;
    then H+G=F-(G-G) by RLVECT_1:29;
    then H+G=F-0.R_Normed_Algebra_of_BoundedFunctions X by RLVECT_1:15;
    then
A4: H+G=F;
    now
      let x be Element of X;
      f.x=h.x + g.x by A1,A4,Th29;
      hence f.x-g.x=h.x;
    end;
    hence for x be Element of X holds h.x = f.x - g.x;
  end;
  hence thesis by A2;
end;
