reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th43:
  for a,b,c,h being Integer st h = (a-b)*(a-c)*(b-c)
  for S being Sierp45FS of a,b,c st a,b,c are_mutually_distinct
  for n being Nat st
  n, numberR(a,b,c) are_congruent_mod 2 &
  n, numberR0(a,b,c) are_congruent_mod 3 &
  for i being Nat st i in dom S holds
    n, S.i are_congruent_mod (Sgm PrimeDivisors>3(h)).i
  holds a+n, b+n, c+n are_mutually_coprime
  proof
    let a,b,c,h be Integer such that
A1: h = (a-b)*(a-c)*(b-c);
    let S be Sierp45FS of a,b,c such that
A2: a,b,c are_mutually_distinct;
    let n be Nat such that
A3: n, numberR(a,b,c) are_congruent_mod 2 and
A4: n, numberR0(a,b,c) are_congruent_mod 3 and
A5: for i being Nat st i in dom S holds
    n, S.i are_congruent_mod (Sgm PrimeDivisors>3(h)).i;
    assume not thesis;
    then per cases;
    suppose not a+n,b+n are_coprime;
      hence thesis by A1,A2,A3,A4,A5,Lm7;
    end;
    suppose not a+n,c+n are_coprime;
      hence thesis by A1,A2,A3,A4,A5,Lm8;
    end;
    suppose not b+n,c+n are_coprime;
      hence thesis by A1,A2,A3,A4,A5,Lm9;
    end;
  end;
