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 Th75:
  <* x,y *> - A = {} iff x in A & y in A
proof
A1: <* x,y *> = <* x *> ^ <* y *> by FINSEQ_1:def 9;
  thus <* x,y *> - A = {} implies x in A & y in A
  proof
    assume <* x,y *> - A = {};
    then rng<* x,y *> c= A by Th66;
    then {x,y} c= A by FINSEQ_2:127;
    hence thesis by ZFMISC_1:32;
  end;
  assume that
A2: x in A and
A3: y in A;
A4: <* y *> - A = {} by A3,Lm7;
  <* x *> - A = {} by A2,Lm7;
  hence <* x,y *> - A = {} ^ {} by A4,A1,Lm11
    .= {};
end;
