reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem
  <* x,y *> - A = <* x,y *> iff not x in A & not y in A
proof
A1: <* x,y *> = <* x *> ^ <* y *> by FINSEQ_1:def 9;
  thus <* x,y *> - A = <* x,y *> implies not x in A & not y in A
  proof
    assume
A2: <* x,y *> - A = <* x,y *>;
    assume not thesis;
    then x in A & y in A or not x in A & y in A or x in A & not y in A;
    then
    <* x,y *> - A = {} or <* x,y *> - A = <* x *> or <* x,y *> - A = <* y
    *> by Th75,Th76,Th78;
    hence thesis by A2,Th33;
  end;
  assume that
A3: not x in A and
A4: not y in A;
A5: <* y *> - A = <* y *> by A4,Lm6;
  <* x *> - A = <* x *> by A3,Lm6;
  hence thesis by A5,A1,Lm11;
end;
