reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;

theorem Th1:
  for F1,F2 being Function of I,A st for x st x in I holds F1/.x =
  F2/.x holds F1 = F2
proof
  let F1,F2 be Function of I,A such that
A1: for x st x in I holds F1/.x = F2/.x;
  now
    let x be object;
    assume
A2: x in I;
    hence F1.x = F1/.x by FUNCT_2:def 13
      .= F2/.x by A1,A2
      .= F2.x by A2,FUNCT_2:def 13;
  end;
  hence thesis by FUNCT_2:12;
end;
