reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th22:
  for x,y being FinSequence of COMPLEX st len x=len y holds (mlt(x
  ,(y*')))*' = mlt(y,(x*'))
proof
  let x,y be FinSequence of COMPLEX;
  assume
A1: len x=len y;
  reconsider x19=x*' as Element of (len (x*'))-tuples_on COMPLEX by FINSEQ_2:92
;
  reconsider y19=y as Element of (len y)-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider y9=y*' as Element of (len (y*'))-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider x9=x as Element of (len x)-tuples_on COMPLEX by FINSEQ_2:92;
A2: len (x*') = len x by COMPLSP2:def 1;
A3: len (y*') = len y by COMPLSP2:def 1;
  then
A4: len mlt(x,(y*')) = len x by A1,FINSEQ_2:72;
A5: len mlt(x,(y*')) = len (y*') by A1,A3,FINSEQ_2:72;
A6: now
    let i be Nat;
    assume that
A7: 1 <= i and
A8: i <= len ((mlt(x,(y*')))*');
A9: i <= len (mlt(x,(y*'))) by A8,COMPLSP2:def 1;
    hence (mlt(x,(y*')))*'.i = ((mlt(x,(y*'))).i)*' by A7,COMPLSP2:def 1
      .= (((x9.i)*((y9).i)))*' by A1,A3,Th18
      .= (x.i)*'*((y*').i)*' by COMPLEX1:35
      .= (x.i)*'*((y*')*'.i) by A5,A7,A9,COMPLSP2:def 1
      .= (x.i)*'*(y.i)
      .= ((x*').i)*(y.i) by A4,A7,A9,COMPLSP2:def 1
      .= mlt(x19,y19).i by A1,A2,Th18
      .= mlt(y,(x*')).i;
  end;
  len (mlt(x,(y*'))) = len ((mlt(x,(y*')))*') by COMPLSP2:def 1;
  then len ((mlt(x,(y*')))*') = len mlt(y,(x*')) by A1,A2,A4,FINSEQ_2:72;
  hence thesis by A6,FINSEQ_1:14;
end;
