
theorem
  for A being free Universal_Algebra for o1,o2 being Element of Operations A
  for p1,p2 being FinSequence st p1 in dom o1 & p2 in dom o2
  holds o1.p1 = o2.p2 implies o1 = o2 & p1 = p2
proof
  let A be free Universal_Algebra;
  let o1,o2 be Element of Operations A;
  consider a1 being object such that
A1: a1 in dom the charact of A and
A2: o1 = (the charact of A).a1 by FUNCT_1:def 3;
  consider a2 being object such that
A3: a2 in dom the charact of A and
A4: o2 = (the charact of A).a2 by FUNCT_1:def 3;
  reconsider a1,a2 as OperSymbol of A by A1,A3;
A5: o1 = Den(a1,A) by A2;
A6: o2 = Den(a2,A) by A4;
  let p1,p2 be FinSequence;
  assume that
A7: p1 in dom o1 and
A8: p2 in dom o2 and
A9: o1.p1 = o2.p2;
  thus thesis by A5,A6,A7,A8,A9,Th36;
end;
