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

theorem Th53:
  for x,y being FinSequence of COMPLEX st len x=len y & 0 < len y
  holds (|(x,y)|)*'=|(x*',y*')|
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;
A4: len (y*') = len y by COMPLSP2:def 1;
  then
A5: len (mlt(x,y*')) = len x by A1,FINSEQ_2:72;
  len (x*') = len x by COMPLSP2:def 1;
  then |(x*',y*')| = Sum (mlt(x*',(y*')*')) by A1,A4,Th37
    .= Sum ((mlt(x,y*'))*') by A1,A4,Th25
    .= (Sum (mlt(x,y*')))*' by A3,A5,Th21
    .= (|(x, y)|)*' by A1,Th37;
  hence thesis;
end;
