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
  sqr mlt(F1,F2) = mlt(sqr F1,sqr F2)
proof
A1: dom mlt(F1,F2) = dom F1 /\ dom F2 by VALUED_1:def 4;
A2: dom mlt(sqr F1,sqr F2) = dom sqr F1 /\ dom sqr F2 by VALUED_1:def 4;
A3: dom sqr F1 = dom F1 by VALUED_1:11;
A4: dom sqr F2 = dom F2 by VALUED_1:11;
    now
      let i be Nat;
      assume i in dom sqr mlt(F1,F2);
      thus (sqr mlt(F1,F2)).i = (mlt(F1,F2).i)^2 by VALUED_1:11
      .= (F1.i*F2.i)^2 by VALUED_1:5
      .= (F1.i)^2*(F2.i)^2
      .= (sqr F1).i*(F2.i)^2 by VALUED_1:11
      .= (sqr F1).i * (sqr F2).i by VALUED_1:11
      .= (mlt(sqr F1,sqr F2)).i by VALUED_1:5;
    end;
    hence thesis by A1,A2,A3,A4,FINSEQ_1:13,VALUED_1:11;
  end;
