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 Th58:
  sqr (r*F) = r^2 * sqr F
proof
A1: dom (r*F) = dom F by VALUED_1:def 5;
A2: dom (r^2 * sqr F) = dom sqr F by VALUED_1:def 5;
A3: dom sqr F = dom F by VALUED_1:11;
    now
      let i be Nat;
      assume i in dom sqr (r*F);
      thus (sqr (r*F)).i = ((r*F).i)^2 by VALUED_1:11
      .= (r*(F.i))^2 by VALUED_1:6
      .= r^2 * (F.i)^2
      .= r^2 * (sqr F).i by VALUED_1:11
      .= (r^2 * sqr F).i by VALUED_1:6;
    end;
    hence thesis by A1,A2,A3,FINSEQ_1:13,VALUED_1:11;
  end;
