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;
reserve X for OrtAfPl;
reserve o9,a9,a19,a29,a39,a49,b9,b19,b29,b39,b49,c9,c19 for Element of X;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1 for Element of the AffinStruct of X;
reserve M9,N9 for Subset of X;
reserve A,M,N for Subset of the AffinStruct of X;

theorem
  the AffinStruct of X is satisfying_Scherungssatz iff X is satisfying_SCH
proof
A1: X is satisfying_SCH implies
    the AffinStruct of X is satisfying_Scherungssatz
  proof
    assume
A2: X is satisfying_SCH;
    now
      let a1,a2,a3,a4,b1,b2,b3,b4,M,N;
      assume that
A3:   M is being_line and
A4:   N is being_line and
A5:   a1 in M and
A6:   a3 in M and
A7:   b1 in M and
A8:   b3 in M and
A9:   a2 in N and
A10:  a4 in N and
A11:  b2 in N and
A12:  b4 in N and
A13:  not a4 in M and
A14:  not a2 in M and
A15:  not b2 in M and
A16:  not b4 in M and
A17:  not a1 in N and
A18:  not a3 in N and
A19:  not b1 in N and
A20:  not b3 in N and
A21:  a3,a2 // b3,b2 and
A22:  a2,a1 // b2,b1 and
A23:  a1,a4 // b1,b4;
      reconsider a19=a1,a29=a2,a39=a3,a49=a4,b19=b1,b29=b2,b39=b3,b49=b4 as
      Element of X;
A24:  a39,a29 // b39,b29 by A21,ANALMETR:36;
A25:  a19,a49 // b19,b49 by A23,ANALMETR:36;
A26:  a29,a19 // b29,b19 by A22,ANALMETR:36;
      reconsider M9=M,N9=N as Subset of X;
A27:  N9 is being_line by A4,ANALMETR:43;
      M9 is being_line by A3,ANALMETR:43;
      then
      a39,a49 // b39,b49 by A2,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17
,A18,A19,A20,A27,A24,A26,A25,CONMETR:def 6;
      hence a3,a4 // b3,b4 by ANALMETR:36;
    end;
    hence thesis;
  end;
  the AffinStruct of X is satisfying_Scherungssatz implies X is satisfying_SCH
  proof
    assume
A28: the AffinStruct of X is satisfying_Scherungssatz;
    now
      let a19,a29,a39,a49,b19,b29,b39,b49,M9,N9;
      assume that
A29:  M9 is being_line and
A30:  N9 is being_line and
A31:  a19 in M9 and
A32:  a39 in M9 and
A33:  b19 in M9 and
A34:  b39 in M9 and
A35:  a29 in N9 and
A36:  a49 in N9 and
A37:  b29 in N9 and
A38:  b49 in N9 and
A39:  not a49 in M9 and
A40:  not a29 in M9 and
A41:  not b29 in M9 and
A42:  not b49 in M9 and
A43:  not a19 in N9 and
A44:  not a39 in N9 and
A45:  not b19 in N9 and
A46:  not b39 in N9 and
A47:  a39,a29 // b39,b29 and
A48:  a29,a19 // b29,b19 and
A49:  a19,a49 // b19,b49;
      reconsider a1=a19,a2=a29,a3=a39,a4=a49,b1=b19,b2=b29,b3=b39,b4=b49 as
      Element of the AffinStruct of X;
A50:  a3,a2 // b3,b2 by A47,ANALMETR:36;
A51:  a1,a4 // b1,b4 by A49,ANALMETR:36;
A52:  a2,a1 // b2,b1 by A48,ANALMETR:36;
      reconsider M=M9,N=N9 as Subset of the AffinStruct of X;
A53:  N is being_line by A30,ANALMETR:43;
      M is being_line by A29,ANALMETR:43;
      then a3,a4 // b3,b4 by A28,A31,A32,A33,A34,A35,A36,A37,A38,A39,A40,A41
,A42,A43,A44,A45,A46,A53,A50,A52,A51;
      hence a39,a49 // b39,b49 by ANALMETR:36;
    end;
    hence thesis by CONMETR:def 6;
  end;
  hence thesis by A1;
end;
