reserve i, j, n for Element of NAT,
  f, g, h, k for FinSequence of REAL,
  M, N for non empty MetrSpace;

theorem
  len f = len h & len g = len k implies sqr (f^g - h^k) = sqr (f-h) ^
  sqr ( g-k)
proof
  assume that
A1: len f = len h and
A2: len g = len k;
A3: len (f^g) = len f + len g by FINSEQ_1:22;
A4: len (h^k) = len h + len k by FINSEQ_1:22;
A5: len sqr (f^g - h^k) = len (f^g - h^k) by Th8
    .= len (f^g) by A1,A2,A3,A4,Th7
    .= len (f-h) + len g by A1,A3,Th7
    .= len (f-h) + len (g-k) by A2,Th7
    .= len sqr (f-h) + len (g-k) by Th8
    .= len sqr (f-h) + len sqr (g-k) by Th8
    .= len (sqr (f-h) ^ sqr (g-k)) by FINSEQ_1:22;
  for i be Nat st 1 <= i & i <= len sqr (f^g - h^k) holds (sqr (f^g - h^k)
  ).i = (sqr (f-h) ^ sqr (g-k)).i
  proof
    let i be Nat;
    assume that
A6: 1 <= i and
A7: i <= len sqr (f^g - h^k);
    i in dom sqr (f^g - h^k) by A6,A7,FINSEQ_3:25;
    then
A8: i in dom (f^g - h^k) by Th8;
    per cases;
    suppose
A9:   i in dom f;
      then
A10:  i in dom (f-h) by A1,Th7;
      then
A11:  i in dom sqr (f-h) by Th8;
A12:  i in dom h by A1,A9,FINSEQ_3:29;
      thus (sqr (f^g - h^k)).i = ((f^g - h^k).i)^2 by VALUED_1:11
        .= ((f^g).i - (h^k).i)^2 by A8,VALUED_1:13
        .= (f.i - (h^k).i)^2 by A9,FINSEQ_1:def 7
        .= (f.i - h.i)^2 by A12,FINSEQ_1:def 7
        .= ((f-h).i)^2 by A10,VALUED_1:13
        .= (sqr (f-h)).i by VALUED_1:11
        .= (sqr (f-h) ^ sqr (g-k)).i by A11,FINSEQ_1:def 7;
    end;
    suppose
      not i in dom f;
      then
A13:  len f < i by A6,FINSEQ_3:25;
      then reconsider j = i - len f as Element of NAT by INT_1:5;
A14:  len (f-h) < i by A1,A13,Th7;
      then
A15:  len sqr (f-h) < i by Th8;
      i <= len (f^g - h^k) by A7,Th8;
      then
A16:  i <= len (f^g) by A1,A2,A3,A4,Th7;
      then i <= len (f-h) + len g by A1,A3,Th7;
      then i <= len (f-h) + len (g-k) by A2,Th7;
      then i - len (f-h) in dom (g-k) by A14,Th4;
      then
A17:  j in dom (g-k) by A1,Th7;
      len f = len (f-h) by A1,Th7;
      then
A18:  j = i - len sqr (f-h) by Th8;
      thus (sqr (f^g - h^k)).i = ((f^g - h^k).i)^2 by VALUED_1:11
        .= ((f^g).i - (h^k).i)^2 by A8,VALUED_1:13
        .= (g.j - (h^k).i)^2 by A13,A16,FINSEQ_1:24
        .= (g.j - k.j)^2 by A1,A2,A3,A4,A13,A16,FINSEQ_1:24
        .= ((g-k).j)^2 by A17,VALUED_1:13
        .= (sqr (g-k)).j by VALUED_1:11
        .= (sqr (f-h) ^ sqr (g-k)).i by A5,A7,A15,A18,FINSEQ_1:24;
    end;
  end;
  hence thesis by A5,FINSEQ_1:14;
end;
