
theorem Th6:
  for OAS being OAffinSpace st OAS is satisfying_DES_1 holds OAS
  is satisfying_DES
proof
  let OAS be OAffinSpace such that
A1: OAS is satisfying_DES_1;
  for o,a,b,c,a1,b1,c1 being Element of OAS st o,a // o,a1 & o,b // o,b1 &
o,c // o,c1 & not o,a,b are_collinear & not o,a,c are_collinear & a,b // a1,b1
& a,c // a1,c1 holds
  b,c // b1,c1
  proof
    let o,a,b,c,a1,b1,c1 be Element of OAS such that
A2: o,a // o,a1 and
A3: o,b // o,b1 and
A4: o,c // o,c1 and
A5: not o,a,b are_collinear and
A6: not o,a,c are_collinear and
A7: a,b // a1,b1 and
A8: a,c // a1,c1;
    consider a2 being Element of OAS such that
A9: Mid a,o,a2 and
A10: o<>a2 by DIRAF:13;
A11: a,o // o,a2 by A9,DIRAF:def 3;
A12: o<>a by A5,DIRAF:31;
    then consider c2 being Element of OAS such that
A13: c,o // o,c2 and
A14: c,a // a2,c2 by A11,ANALOAF:def 5;
A15: c2,a2 // a,c by A14,DIRAF:2;
A16: c2,o // o,c by A13,DIRAF:2;
    then Mid c2,o,c by DIRAF:def 3;
    then
A17: c2,o,c are_collinear by DIRAF:28;
    a,o,a2 are_collinear by A9,DIRAF:28;
    then
A18: o,a2,a are_collinear by DIRAF:30;
A19: o<>c2
    proof
      assume o=c2;
      then o,a2 // a,c by A14,DIRAF:2;
      then o,a2 '||' a,c by DIRAF:def 4;
      then o,a2,o are_collinear & o,a2,c are_collinear by A10,A18,DIRAF:31,33;
      hence contradiction by A6,A10,A18,DIRAF:32;
    end;
A20: not o,a2,c2 are_collinear
    proof
A21:  c2,o,o are_collinear by DIRAF:31;
A22:  o,a2,o are_collinear by DIRAF:31;
      assume o,a2,c2 are_collinear;
      then c2,o,a are_collinear by A10,A18,A22,DIRAF:32;
      hence contradiction by A6,A17,A19,A21,DIRAF:32;
    end;
    consider b2 being Element of OAS such that
A23: b,o // o,b2 and
A24: b,a // a2,b2 by A12,A11,ANALOAF:def 5;
A25: b2,a2 // a,b by A24,DIRAF:2;
    a<>b by A5,DIRAF:31;
    then b2,a2 // a1,b1 by A7,A25,DIRAF:3;
    then
A26: a2,b2 // b1,a1 by DIRAF:2;
    o<>c by A6,DIRAF:31;
    then
A27: c2,o // o,c1 by A4,A16,DIRAF:3;
A28: a,c // c2,a2 by A14,ANALOAF:def 5;
A29: b2,o // o,b by A23,DIRAF:2;
    then Mid b2,o,b by DIRAF:def 3;
    then
A30: b2,o,b are_collinear by DIRAF:28;
A31: o<>b2
    proof
      assume o=b2;
      then o,a2 // a,b by A24,DIRAF:2;
      then o,a2 '||' a,b by DIRAF:def 4;
      then o,a2,o are_collinear & o,a2,b are_collinear by A10,A18,DIRAF:31,33;
      hence contradiction by A5,A10,A18,DIRAF:32;
    end;
A32: not o,a2,b2 are_collinear
    proof
A33:  b2,o,o are_collinear by DIRAF:31;
A34:  o,a2,o are_collinear by DIRAF:31;
      assume o,a2,b2 are_collinear;
      then b2,o,a are_collinear by A10,A18,A34,DIRAF:32;
      hence contradiction by A5,A30,A31,A33,DIRAF:32;
    end;
A35: now
      b2,a2 // a,b by A24,DIRAF:2;
      then
A36:  b2,a2 '||' a,b by DIRAF:def 4;
      assume
A37:  c2=b2;
      then
A38:  o,b2,c are_collinear by A17,DIRAF:30;
      c2,a2 // a,c by A14,DIRAF:2;
      then
A39:  b2,a2 '||' a,c by A37,DIRAF:def 4;
      ( not o,b2,a2 are_collinear)& o,b2,b are_collinear by A30,A32,DIRAF:30;
      then b=c by A18,A38,A36,A39,PASCH:4;
      hence thesis by DIRAF:4;
    end;
    a2,o // o,a by A11,DIRAF:2;
    then
A40: a2,o // o,a1 by A2,A12,DIRAF:3;
    a<>c by A6,DIRAF:31;
    then c2,a2 // a1,c1 by A8,A15,DIRAF:3;
    then
A41: a2,c2 // c1,a1 by DIRAF:2;
    o<>b by A5,DIRAF:31;
    then b2,o // o,b1 by A3,A29,DIRAF:3;
    then
A42: c2,b2 // b1,c1 by A1,A40,A27,A41,A26,A32,A20;
    a,b // b2,a2 by A24,ANALOAF:def 5;
    then b,c // c2,b2 by A1,A5,A6,A11,A23,A13,A28;
    hence thesis by A42,A35,DIRAF:3;
  end;
  hence thesis;
end;
