reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th25:
  sqr Base_FinSeq(n,i) = Base_FinSeq(n,i)
proof
A1: dom (sqrreal * Base_FinSeq(n,i)) = (Base_FinSeq(n,i))"(dom sqrreal) by
RELAT_1:147;
A2: rng (Base_FinSeq(n,i)) c= REAL;
A3: for x being object st x in dom Base_FinSeq(n,i) holds (sqrreal * (
  Base_FinSeq(n,i) qua Function)).x=(Base_FinSeq(n,i)).x
  proof
    let x be object;
    assume
A4: x in dom Base_FinSeq(n,i);
    then reconsider nx=x as Element of NAT;
A5: (sqrreal * ( Base_FinSeq(n,i) qua Function)).x =sqrreal.((Base_FinSeq
    (n,i)).x) by A4,FUNCT_1:13;
A6: x in Seg len (Base_FinSeq(n,i)) by A4,FINSEQ_1:def 3;
    then
A7: 1<=nx by FINSEQ_1:1;
    len (Base_FinSeq(n,i))=n by MATRIXR2:74;
    then
A8: nx<=n by A6,FINSEQ_1:1;
    per cases;
    suppose
A9:   nx=i;
      hence
      (sqrreal * ( Base_FinSeq(n,i) qua Function)).x =sqrreal.1 by A7,A8,A5,
MATRIXR2:75
        .=1^2 by RVSUM_1:def 2
        .= (Base_FinSeq(n,i)).x by A7,A8,A9,MATRIXR2:75;
    end;
    suppose
A10:  nx<>i;
      hence
      (sqrreal * ( Base_FinSeq(n,i) qua Function)).x =sqrreal.0 by A7,A8,A5,
MATRIXR2:76
        .=0^2 by RVSUM_1:def 2
        .= (Base_FinSeq(n,i)).x by A7,A8,A10,MATRIXR2:76;
    end;
  end;
  (Base_FinSeq(n,i))"(dom sqrreal) =(Base_FinSeq(n,i))"REAL by FUNCT_2:def 1
    .=dom (Base_FinSeq(n,i)) by A2,Th1;
  hence thesis by A1,A3,FUNCT_1:2;
end;
