reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;
reserve f1,f2,f3 for FinSequence of REAL;
reserve n1,n2,m1,m2 for Nat;

theorem Th15:
  n1 in NatDivisors n & m1 in NatDivisors m & n2 in NatDivisors n
  & m2 in NatDivisors m & n,m are_coprime & n1*m1 = n2*m2
  implies n1 = n2 & m1 = m2
proof
  assume
A1: n1 in NatDivisors n;
  then
A2: n1 divides n by MOEBIUS1:39;
  assume
A3: m1 in NatDivisors m;
  then
A4: m1 divides m by MOEBIUS1:39;
  assume
A5: n2 in NatDivisors n;
  then
A6: n2 divides n by MOEBIUS1:39;
  assume
A7: m2 in NatDivisors m;
  assume
A8: n,m are_coprime;
  assume
A9: n1*m1=n2*m2;
A10: m2 divides m by A7,MOEBIUS1:39;
A11: now
    reconsider m19=m1,m29=m2 as non zero Nat by A3,A7,MOEBIUS1:39;
A12: n gcd m > 0 by A8,INT_2:def 3;
    reconsider n19=n1,n29=n2 as non zero Nat by A1,A5,MOEBIUS1:39;
    assume n1<>n2;
    then consider p be Element of NAT such that
A13: p is prime and
A14: p |-count n19 <> p |-count n29 by NAT_4:21;
    reconsider p as Prime by A13;
    (p |-count n19) + (p |-count m19) = p |-count (n19*m19) by NAT_3:28
      .= (p |-count n29) + (p |-count m29) by A9,NAT_3:28;
    then p |-count m19 <> 0 or p |-count m29 <> 0 by A14;
    then p divides m19 or p divides m29 by MOEBIUS1:6;
    then
A15: p divides m by A4,A10,INT_2:9;
    p |-count n19 <> 0 or p |-count n29 <> 0 by A14;
    then p divides n19 or p divides n29 by MOEBIUS1:6;
    then p divides n by A2,A6,INT_2:9;
    then p divides n gcd m by A15,INT_2:def 2;
    then
A16: p <= n gcd m by A12,INT_2:27;
    p > 1 by INT_2:def 4;
    hence contradiction by A8,A16,INT_2:def 3;
  end;
A17: 0 < n2 by A5,MOEBIUS1:39;
  assume not (n1=n2 & m1=m2);
  hence contradiction by A17,A9,A11,XCMPLX_1:5;
end;
