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;

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;
