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

theorem Th52:
  for x,y being FinSequence of COMPLEX st len x=len y & 0 < len y
  holds |(x,y)|=(|(y,x)|)*'
proof
  let x,y be FinSequence of COMPLEX;
  assume that
A1: len x=len y and
A2: 0 < len y;
A3: 0+1 <= len x by A1,A2,NAT_1:8;
  len (x*') = len (x) by COMPLSP2:def 1;
  then
A4: len (mlt(y,(x*'))) = len y by A1,FINSEQ_2:72;
  (|(y,x)|)*' = (Sum (mlt(y,(x*'))))*' by A1,Th37
    .= Sum (mlt(y,(x*'))*') by A1,A3,A4,Th21
    .= Sum (mlt(x,(y*'))) by A1,Th22
    .= |(x, y)| by A1,Th37;
  hence thesis;
end;
