
theorem Th9:
  for x be Complex holds |.<*x*>.| = <*|.x.|*>
proof
  let x be Complex;
  0+1 in Seg (0+1) by FINSEQ_1:4;
  then
A1: 1 in dom <*x*> by FINSEQ_1:38;
A2: len |.<*x*>.| = len <*x*> by Def2
    .= 1 by FINSEQ_1:39;
  then
A3: dom |.<*x*>.| = Seg 1 by FINSEQ_1:def 3;
A5: now
    let n be Nat;
    assume n in dom |.<*x*>.|;
    then
A6: n = 1 by A3,FINSEQ_1:2,TARSKI:def 1;
    hence |.<*x*>.|.n = |.<*x*>.1 .| by A1,Def2
      .= |.x.|
      .= <*|.x.|*>.n by A6;
  end;
  len <*|.x.|*> = 1 by FINSEQ_1:39;
  hence thesis by A2,A5,FINSEQ_2:9;
end;
