reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

theorem Th69:
  sqr(F1 - F2) = sqr F1 - 2*mlt(F1,F2) + sqr F2
proof
  thus sqr(F1 - F2) = sqr F1 + 2*mlt(F1,-F2) + sqr -F2 by Th68
    .= sqr F1 + 2*mlt(F1,-F2) + sqr F2 by Th57
    .= sqr F1 + 2*((-1)*mlt(F1,F2)) + sqr F2 by RFUNCT_1:12
    .= sqr F1 + ((-1)*2)*mlt(F1,F2) + sqr F2 by RFUNCT_1:17
    .= sqr F1 - 2*mlt(F1,F2) + sqr F2 by RFUNCT_1:17;
end;
