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;

theorem
  x 'in' y = z 'in' t implies x = z & y = t
proof
A1: <*1*>^<*x*>^<*y*> = <*1*>^(<*x*>^<*y*>) & <*1*>^<*z*>^<*t*> = <*1*>^(<*z
  *>^ <*t*>) by FINSEQ_1:32;
A2: <*x,y*>.1 = x & <*x,y*>.2 = y;
A3: <*x*>^<*y*> = <*x,y*> & <*z*>^<*t*> = <*z,t*> by FINSEQ_1:def 9;
A4: <*z,t*>.1 = z & <*z,t*>.2 = t;
  assume x 'in' y = z 'in' t;
  hence thesis by A1,A2,A4,A3,FINSEQ_1:33;
end;
