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 Th68:
  sqr(F1 + F2) = sqr F1 + 2*mlt(F1,F2) + sqr F2
proof
A1: dom sqr(F1 + F2) = dom (F1+F2) by VALUED_1:11;
A2: dom(F1+F2) = dom F1 /\ dom F2 by VALUED_1:def 1;
A3: dom(sqr F1 + 2*mlt(F1,F2))
    = dom(sqr F1) /\ dom (2*mlt(F1,F2)) by VALUED_1:def 1
    .= dom F1 /\ dom (2*mlt(F1,F2)) by VALUED_1:11
    .= dom F1 /\ dom (mlt(F1,F2)) by VALUED_1:def 5
    .= dom F1 /\ (dom F1 /\ dom F2) by VALUED_1:def 4
    .= dom F1 /\ dom F1 /\ dom F2 by XBOOLE_1:16
    .= dom F1 /\ dom F2;
then A4: dom(sqr F1 + 2*mlt(F1,F2) + sqr F2)
    = dom F1 /\ dom F2 /\ dom sqr F2 by VALUED_1:def 1
    .= dom F1 /\ dom F2 /\ dom F2 by VALUED_1:11
    .= dom F1 /\ (dom F2 /\ dom F2) by XBOOLE_1:16;
  now
    let j be Nat;
    assume
A5: j in dom sqr(F1+F2);
    set r1r2 = (F1 + F2).j, r1 = F1.j, r2 = F2.j, r192 = (sqr F1).j,
    r292 = (sqr F2).j, r1r2a = mlt(F1,F2).j, 2r1r2 = (2*mlt(F1,F2)).j,
    r1922r1r2 = (sqr F1 + 2*mlt(F1,F2)).j;
    thus (sqr(F1 + F2)).j = r1r2^2 by VALUED_1:11
      .= (r1 + r2)^2 by A1,A5,VALUED_1:def 1
      .= r1^2+2*r1*r2+r2^2
      .= r192+2*(r1*r2)+r2^2 by VALUED_1:11
      .= r192+2*(r1*r2)+r292 by VALUED_1:11
      .= r192+2*(r1r2a)+r292 by VALUED_1:5
      .= r192+2r1r2+r292 by VALUED_1:6
      .= r1922r1r2+r292 by A1,A2,A3,A5,VALUED_1:def 1
      .= (sqr F1 + 2*mlt(F1,F2) + sqr F2).j by A1,A2,A4,A5,VALUED_1:def 1;
  end;
  hence thesis by A2,A4,FINSEQ_1:13,VALUED_1:11;
end;
