
theorem LemmaFor52:
  for m,k,l being Nat st k <> l & 1 <= k <= m & 1 <= l <= m holds
    m! * k + 1, m! * l + 1 are_coprime
  proof
    let m,k,l be Nat;
    assume
S1: k <> l & 1 <= k <= m & 1 <= l <= m;
    assume not m! * k + 1, m! * l + 1 are_coprime; then
    consider d being non zero Nat such that
A1: d <> 1 & d divides m! * k + 1 & d divides m! * l + 1 by MOEBIUS2:5;
    per cases by S1,XXREAL_0:1;
    suppose
      k < l; then
b1:   l - k > k - k by XREAL_1:14;
a1:   d divides l * (m! * k + 1) by A1,NAT_D:9;
      d divides k * (m! * l + 1) by A1,NAT_D:9; then
A2:   d divides (l * (m! * k + 1) - k * (m! * l + 1)) by a1,PREPOWER:94;
A3:   l - k <= m - k by S1,XREAL_1:13;
      m - k < m by S1,XREAL_1:44; then
A5:   l - k < m by A3,XXREAL_0:2;
      d <= l - k by b1,A2,NAT_D:7; then
      d <= m by A5,XXREAL_0:2; then
      d divides m! * k by NAT_D:9,NEWTON:41;
      hence thesis by A1,WSIERP_1:15,NAT_D:10;
    end;
    suppose
      k > l; then
b1:   k - l > l - l by XREAL_1:14;
a1:   d divides l * (m! * k + 1) by A1,NAT_D:9;
      d divides k * (m! * l + 1) by A1,NAT_D:9; then
A2:   d divides (k * (m! * l + 1) - l * (m! * k + 1)) by a1,PREPOWER:94;
A3:   k - l <= m - l by S1,XREAL_1:13;
      m - l < m by S1,XREAL_1:44; then
A5:   k - l < m by A3,XXREAL_0:2;
      d <= k - l by b1,A2,NAT_D:7; then
      d <= m by A5,XXREAL_0:2; then
      d divides m! * k by NAT_D:9,NEWTON:41;
      hence thesis by A1,WSIERP_1:15,NAT_D:10;
    end;
  end;
