reserve X for AffinPlane;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1,c2,d,d1,d2, d3,d4,d5,e1,e2,x,y,z
  for Element of X;
reserve Y,Z,M,N,A,K,C for Subset of X;

theorem
  X is satisfying_PPAP iff X is satisfying_indirect_Scherungssatz
proof
A1: X is satisfying_PPAP implies X is satisfying_indirect_Scherungssatz
  proof
    assume
A2: X is satisfying_PPAP;
    then X is satisfying_pap by AFF_2:10;
    then
A3: X is satisfying_minor_indirect_Scherungssatz by Th9;
    X is Pappian by A2,AFF_2:10;
    then X is satisfying_major_indirect_Scherungssatz by Th10;
    hence thesis by A3,Th1;
  end;
  X is satisfying_indirect_Scherungssatz implies X is satisfying_PPAP
  proof
    assume
A4: X is satisfying_indirect_Scherungssatz;
    then X is satisfying_major_indirect_Scherungssatz;
    then
A5: X is Pappian by Th10;
    X is satisfying_minor_indirect_Scherungssatz by A4,Th1;
    then X is satisfying_pap by Th9;
    hence thesis by A5,AFF_2:10;
  end;
  hence thesis by A1;
end;
