theorem Th12:
  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;
  reconsider f1=F, g1=G, h1=H as VECTOR of RAlgebra X by TARSKI:def 3;
A2: R_Algebra_of_BoundedFunctions X is Subalgebra of RAlgebra X by Th6;
  hereby
    assume
A3: H = F+G;
    let x be Element of X;
    h1=f1+g1 by A2,A3,Th8;
    hence h.x = f.x+g.x by A1,FUNCSDOM:1;
  end;
  assume for x be Element of X holds h.x = f.x+g.x;
  then h1=f1+g1 by A1,FUNCSDOM:1;
  hence thesis by A2,Th8;
end;
