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 = <* y *> & x <> y implies x in A & not y in A
proof
  assume that
A1: <* x,y *> - A = <* y *> and
A2: x <> y;
  assume
A3: not thesis;
A4: y in A implies <* y *> - A = {} by Lm7;
A5: not x in A implies <* x *> - A = <* x *> by Lm6;
A6: not y in A implies <* y *> - A = <* y *> by Lm6;
A7: x in A implies <* x *> - A = {} by Lm7;
A8: <* x *>.1 = x;
  <* y *> = (<* x *> ^ <* y *>) - A by A1,FINSEQ_1:def 9
    .= (<* x *> - A) ^ (<* y *> - A) by Lm11;
  then <* y *> = {} or <* x *> = <* y *> or <* y *> = <* x,y *>
    by A7,A5,A4,A6,A3,FINSEQ_1:34,def 9;
  hence thesis by A2,A8,Th33,FINSEQ_1:40;
end;
