
theorem
  for x,y,z be Complex holds |.<*x,y,z*>.| = <*|.x.|,|.y.|,|.z.|*>
proof
  let x,y,z be Complex;
A1: len |.<*x,y,z*>.| = len <*x,y,z*> by Def2
    .= 3 by FINSEQ_1:45;
  then
A2: dom |.<*x,y,z*>.| = Seg 3 by FINSEQ_1:def 3;
A3: now
    let n be Nat;
    assume
A4: n in dom |.<*x,y,z*>.|;
    per cases by A2,A4,ENUMSET1:def 1,FINSEQ_3:1;
    suppose
A5:   n = 1;
A6:   1 in dom <*x,y,z*> by FINSEQ_1:81;
      |.<*x,y,z*>.|.1 = |.<*x,y,z*>.1 .| by A6,Def2;
      then |.<*x,y,z*>.|.1 = |.x.|;
      hence |.<*x,y,z*>.|.n = <*|.x.|,|.y.|,|.z.|*>.n by A5;
    end;
    suppose
A7:   n = 2;
A8:   2 in dom <*x,y,z*> by FINSEQ_1:81;
      |.<*x,y,z*>.|.2 = |.<*x,y,z*>.2 .| by A8,Def2;
      then |.<*x,y,z*>.|.2 = |.y.|;
      hence |.<*x,y,z*>.|.n = <*|.x.|,|.y.|,|.z.|*>.n by A7;
    end;
    suppose
A9:   n = 3;
A10:  3 in dom <*x,y,z*> by FINSEQ_1:81;
      |.<*x,y,z*>.|.3 = |.<*x,y,z*>.3 .| by A10,Def2;
      then |.<*x,y,z*>.|.3 = |.z.|;
      hence |.<*x,y,z*>.|.n = <*|.x.|,|.y.|,|.z.|*>.n by A9;
    end;
  end;
  len <*|.x.|,|.y.|,|.z.|*> = 3 by FINSEQ_1:45;
  hence thesis by A1,A3,FINSEQ_2:9;
end;
