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 Th13:
  f=F & g=G implies ( G = a*F iff for x be Element of X holds g.x = a*f.x )
proof
  assume
A1: f=F & g=G;
  reconsider f1=F, g1=G 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: G = a*F;
    let x be Element of X;
    g1=a*f1 by A2,A3,Th8;
    hence g.x=a*f.x by A1,FUNCSDOM:4;
  end;
  assume for x be Element of X holds g.x=a*f.x;
  then g1=a*f1 by A1,FUNCSDOM:4;
  hence thesis by A2,Th8;
end;
