reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;

theorem Th15:
  for x,y holds (x '=' y).1 = 0 & (x 'in' y ).1 = 1
proof
  let x,y;
  thus (x '=' y).1 = (<*0*>^(<*x*>^<*y*>)).1 by FINSEQ_1:32
    .= 0 by FINSEQ_1:41;
  thus (x 'in' y).1 = (<*1*>^(<*x*>^<*y*>)).1 by FINSEQ_1:32
    .= 1 by FINSEQ_1:41;
end;
