
theorem Th19:
  for m, n be non zero Nat
  for f be PartFunc of REAL m,REAL n
  for x be Element of REAL m
  holds
    f is_differentiable_in x
      iff
    for i be Nat st 1 <= i <= n
    holds
    ex fi be PartFunc of REAL m,REAL
    st fi = proj(i,n) * f
     & fi is_differentiable_in x
proof
  let m, n be non zero Nat;
  let f be PartFunc of REAL m,REAL n;
  let x be Element of REAL m;

  reconsider y = x as Point of REAL-NS m by REAL_NS1:def 4;

  A2: for i be Nat holds <>* (proj(i,n) * f) = Proj(i,n) * f
  proof
    let i be Nat;
    reconsider fi = proj(i,n) * f as PartFunc of REAL m,REAL;
    Proj(i,n) * f
    = ((proj(1,1) qua Function)" * proj(i,n)) * f by PDIFF_1:11
    .= <>* fi by RELAT_1:36;
    hence thesis;
  end;

  hereby
    assume
    A3: f is_differentiable_in x;

    thus
    for i be Nat st 1 <= i <= n
    holds
    ex fi be PartFunc of REAL m,REAL
    st fi = proj(i,n) * f
      & fi is_differentiable_in x
    proof
      let i be Nat;
      assume 1 <= i <= n;
      then consider Fi be PartFunc of REAL m,REAL 1 such that
      A4: Fi = Proj(i,n) * f
        & Fi is_differentiable_in x by A3,Th18;

      reconsider fi = proj(i,n) * f as PartFunc of REAL m,REAL;
      take fi;
      thus fi = proj(i,n) * f;
      <>* fi = Fi by A4,A2;
      hence fi is_differentiable_in x by A4,PDIFF_7:def 1;
    end;
  end;

  assume
  A5: for i be Nat st 1 <= i <= n
      holds
      ex fi be PartFunc of REAL m,REAL
      st fi = proj(i,n) * f
        & fi is_differentiable_in x;

  for i be Nat st 1 <= i <= n
  holds
  ex Fi be PartFunc of REAL m,REAL 1
  st Fi = Proj(i,n) * f
    & Fi is_differentiable_in x
  proof
    let i be Nat;
    assume 1 <= i <= n;
    then consider fi be PartFunc of REAL m,REAL such that
    A6: fi = proj(i,n) * f
      & fi is_differentiable_in x by A5;

    reconsider Fi = Proj(i,n) * f
      as PartFunc of REAL m,REAL 1 by REAL_NS1:def 4;

    take Fi;
    thus Fi = Proj(i,n) * f;

    Fi = <>* fi by A2,A6;
    hence Fi is_differentiable_in x by A6,PDIFF_7:def 1;
  end;
  hence f is_differentiable_in x by Th18;
end;
