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 Th57:
  sqr -F = sqr F
proof
A1: dom sqr -F = dom -F by VALUED_1:11
    .= dom F by VALUED_1:8;
A2: dom sqr F = dom F by VALUED_1:11;
  now
    let j be Nat;
    assume j in dom sqr -F;
    set r = F.j, r9 = (-F).j;
    thus (sqr -F).j = (r9)^2 by VALUED_1:11
      .= (-r)^2 by VALUED_1:8
      .= r^2
      .= (sqr F).j by VALUED_1:11;
  end;
  hence thesis by A1,A2,FINSEQ_1:13;
end;
