reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem Th5:
  for x1,x2 being FinSequence of L,i being Nat st len x1=len x2 & 1
  <=i & i<=len x1 holds (x1+x2).i=x1/.i + x2/.i & (x1-x2).i=x1/.i - x2/.i
proof
  let x1,x2 be FinSequence of L,i be Nat;
  assume that
A1: len x1=len x2 and
A2: 1<=i & i<=len x1;
  reconsider x10=x1,x20=x2 as Element of (len x1)-tuples_on the carrier of L
  by A1,Th1;
A3: x10/.i=x10.i & x20/.i=x20.i by A1,A2,FINSEQ_4:15;
  i in Seg len x1 by A2,FINSEQ_1:1;
  then i in Seg len (x1+x2) by A1,Th2;
  then i in dom (x1+x2) by FINSEQ_1:def 3;
  hence (x1+x2).i=x1/.i + x2/.i by A3,FVSUM_1:17;
  i in Seg len x1 by A2,FINSEQ_1:1;
  then i in Seg len (x1-x2) by A1,Th3;
  then i in dom (x1-x2) by FINSEQ_1:def 3;
  hence thesis by A3,FVSUM_1:32;
end;
