reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th17:
  for x1,x2 being FinSequence of COMPLEX st len x1=len x2 holds
  (x1 - x2)*' = x1*' - x2*'
proof
  let x1,x2 be FinSequence of COMPLEX;
  reconsider x9=x1 as Element of (len x1)-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider y9=x2 as Element of (len x2)-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider x3=x1*' as Element of (len (x1*'))-tuples_on COMPLEX by
FINSEQ_2:92;
  reconsider x4=x2*' as Element of (len (x2*'))-tuples_on COMPLEX by
FINSEQ_2:92;
  assume
A1: len x1=len x2; then
A2: len (x1 - x2) = len x1 by Th7;
A3: len x1=len (x1*') & len x2=len (x2*') by Def1;
A4: now
    let i be Nat;
A5: i in NAT by ORDINAL1:def 12;
    assume that
A6: 1 <= i and
A7: i <= len ((x1 - x2)*');
A8: i <= len (x1 - x2) by A7,Def1;
    hence (x1 - x2)*'.i = (((x1 - x2).i)*') by A6,Def1
      .= (x9.i - y9.i)*' by A1,A5,Th16
      .= (x1.i)*'-(x2.i)*' by COMPLEX1:34
      .= ((x1*').i)-(x2.i)*' by A2,A6,A8,Def1
      .= ((x1*').i)-((x2*').i) by A1,A2,A6,A8,Def1
      .= (x3-x4).i by A1,A3,A5,Th16;
  end;
  len (x1*' - x2*') = len x1 by A1,A3,Th7;
  hence thesis by A4,A2,Def1;
end;
