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

theorem Th25:
  for x,y being FinSequence of COMPLEX st len x=len y holds (mlt(x
  ,y))*' = mlt(x*',y*')
proof
  let x,y be FinSequence of COMPLEX;
A1: len ((mlt(x,y))*') = len (mlt(x,y)) by COMPLSP2:def 1;
A2: len (x*') = len x by COMPLSP2:def 1;
  assume
A3: len x=len y;
A4: for i be Nat st 1 <=i & i <= len ((mlt(x,y))*') holds (mlt(x,(y)))*'.i=
  mlt(x*',y*').i
  proof
    let i be Nat;
    (mlt(x,y))*'.i = ((mlt(y,(x*')*')*').i)
      .= (mlt(x*',y*').i) by A3,A2,Th22;
    hence thesis;
  end;
  len (y*') = len y by COMPLSP2:def 1;
  then len (mlt(x*',y*')) = len (x*') by A3,A2,FINSEQ_2:72;
  then len ((mlt(x,y))*') = len (mlt(x*',y*')) by A3,A1,A2,FINSEQ_2:72;
  hence thesis by A4,FINSEQ_1:14;
end;
