
theorem
  for x,y be Complex holds |.<*x,y*>.| = <*|.x.|,|.y.|*>
proof
  let x,y be Complex;
A1: len |.<*x,y*>.| = len <*x,y*> by Def2
    .= 2 by FINSEQ_1:44;
  then
A2: dom |.<*x,y*>.| = Seg 2 by FINSEQ_1:def 3;
A3: now
    let n be Nat;
    assume
A4: n in dom |.<*x,y*>.|;
    per cases by A2,A4,FINSEQ_1:2,TARSKI:def 2;
    suppose
A5:   n = 1;
      then
A6:   1 in dom <*x,y*> by A2,A4,FINSEQ_1:89;
      |.<*x,y*>.|.1 = |.<*x,y*>.1 .| by A6,Def2;
      then |.<*x,y*>.|.1 = |.x.|;
      hence |.<*x,y*>.|.n = <*|.x.|,|.y.|*>.n by A5;
    end;
    suppose
A7:   n = 2;
      then
A8:   2 in dom <*x,y*> by A2,A4,FINSEQ_1:89;
      |.<*x,y*>.|.2 = |.<*x,y*>.2 .| by A8,Def2;
      then |.<*x,y*>.|.2 = |.y.|;
      hence |.<*x,y*>.|.n = <*|.x.|,|.y.|*>.n by A7;
    end;
  end;
  len <*|.x.|,|.y.|*> = 2 by FINSEQ_1:44;
  hence thesis by A1,A3,FINSEQ_2:9;
end;
