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

theorem Th20:
  for X be non empty closed_interval Subset of REAL
  for Y be RealNormSpace
  for f,g,h be Point of R_NormSpace_of_ContinuousFunctions(X,Y)
  for f9,g9,h9 be continuous PartFunc of REAL,Y
  st f9=f & g9=g & h9=h & dom f9=X & dom g9=X & dom h9=X
  holds (h = f-g iff for x be Element of X holds h9/.x = f9/.x - g9/.x )
proof
  let X be non empty closed_interval Subset of REAL;
  let Y be RealNormSpace;
  let f,g,h be Point of R_NormSpace_of_ContinuousFunctions(X,Y);
  let f9,g9,h9 be continuous PartFunc of REAL,Y such that
A1:  f9=f & g9=g & h9=h & dom f9=X & dom g9=X & dom h9=X;
A2: now
    assume
A3: for x be Element of X holds h9/.x = f9/.x - g9/.x;
    now
      let x be Element of X;
      h9/.x = f9/.x - g9/.x by A3;
      then h9/.x + g9/.x= f9/.x - (g9/.x- g9/.x) by RLVECT_1:29;
      then h9/.x + g9/.x= f9/.x - 0.Y by RLVECT_1:15;
      hence h9/.x + g9/.x= f9/.x;
    end;
    then f=h+g by A1,Th15;
    then f-g=h+(g-g) by RLVECT_1:def 3;
    then f-g=h+0.R_NormSpace_of_ContinuousFunctions(X,Y) by RLVECT_1:5;
    hence f-g=h;
  end;
  now
    assume h=f-g;
    then h+g=f-(g-g) by RLVECT_1:29;
    then
A4: h+g=f-0.R_NormSpace_of_ContinuousFunctions(X,Y) by RLVECT_1:5;
    now
      let x be Element of X;
      f9/.x=h9/.x + g9/.x by A1,A4,Th15;
      then f9/.x-g9/.x=h9/.x + (g9/.x-g9/.x) by RLVECT_1:def 3;
      then f9/.x-g9/.x=h9/.x + 0.Y by RLVECT_1:15;
      hence f9/.x-g9/.x=h9/.x;
    end;
    hence for x be Element of X holds h9/.x = f9/.x - g9/.x;
  end;
  hence thesis by A2;
end;
