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 Th2:
  X is satisfying_Scherungssatz iff X is
  satisfying_minor_Scherungssatz & X is satisfying_major_Scherungssatz
proof
A1: X is satisfying_Scherungssatz implies X is satisfying_minor_Scherungssatz
  proof
    assume
A2: X is satisfying_Scherungssatz;
    now
      let a1,a2,a3,a4,b1,b2,b3,b4,M,N;
      assume that
A3:   M // N and
A4:   a1 in M and
A5:   a3 in M and
A6:   b1 in M and
A7:   b3 in M and
A8:   a2 in N and
A9:   a4 in N and
A10:  b2 in N and
A11:  b4 in N and
A12:  not a4 in M and
A13:  not a2 in M and
A14:  not b2 in M and
A15:  not b4 in M and
A16:  not a1 in N and
A17:  not a3 in N and
A18:  not b1 in N and
A19:  not b3 in N and
A20:  a3,a2 // b3,b2 and
A21:  a2,a1 // b2,b1 and
A22:  a1,a4 // b1,b4;
A23:  N is being_line by A3,AFF_1:36;
      M is being_line by A3,AFF_1:36;
      hence a3,a4 // b3,b4 by A2,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,A16
,A17,A18,A19,A20,A21,A22,A23;
    end;
    hence thesis;
  end;
A24: X is satisfying_minor_Scherungssatz & X is
  satisfying_major_Scherungssatz implies X is satisfying_Scherungssatz
  proof
    assume that
A25: X is satisfying_minor_Scherungssatz and
A26: X is satisfying_major_Scherungssatz;
    now
      let a1,a2,a3,a4,b1,b2,b3,b4,M,N;
      assume that
A27:  M is being_line and
A28:  N is being_line and
A29:  a1 in M and
A30:  a3 in M and
A31:  b1 in M and
A32:  b3 in M and
A33:  a2 in N and
A34:  a4 in N and
A35:  b2 in N and
A36:  b4 in N and
A37:  not a4 in M and
A38:  not a2 in M and
A39:  not b2 in M and
A40:  not b4 in M and
A41:  not a1 in N and
A42:  not a3 in N and
A43:  not b1 in N and
A44:  not b3 in N and
A45:  a3,a2 // b3,b2 and
A46:  a2,a1 // b2,b1 and
A47:  a1,a4 // b1,b4;
      now
        assume not M // N;
        then ex o st o in M & o in N by A27,A28,AFF_1:58;
        hence a3,a4 // b3,b4 by A26,A27,A28,A29,A30,A31,A32,A33,A34,A35,A36,A37
,A38,A39,A40,A41,A42,A43,A44,A45,A46,A47;
      end;
      hence
      a3,a4 // b3,b4 by A25,A29,A30,A31,A32,A33,A34,A35,A36,A37,A38,A39,A40,A41
,A42,A43,A44,A45,A46,A47;
    end;
    hence thesis;
  end;
  thus thesis by A1,A24;
end;
