reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th21:
  for X be non empty closed_interval Subset of REAL for Y be RealNormSpace,
    f,g be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
    f1,g1 be Point of R_NormSpace_of_BoundedFunctions(X,Y)
      st f1=f & g1=g
      holds f-g = f1-g1
proof
  let X be non empty closed_interval Subset of REAL,Y be RealNormSpace,
   f,g  be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
  f1,g1 be Point of R_NormSpace_of_BoundedFunctions(X,Y);
  assume A1: f1=f & g1=g;
A2: -g1 = (-1)*g1 by RLVECT_1:16
       .= (-1)*g by A1,Th19
       .= -g by RLVECT_1:16;
  thus f-g = f1 - g1 by A1,A2,Th18;
end;
