
theorem Th11:
  for V being RealLinearSpace, OAS being OAffinSpace st OAS =
  OASpace(V) holds OAS is satisfying_DES_1
proof
  let V be RealLinearSpace,OAS be OAffinSpace such that
A1: OAS = OASpace(V);
  for o,a,b,c,a1,b1,c1 being Element of OAS st a,o // o,a1 & b,o // o,b1 &
c,o // o,c1 & not o,a,b are_collinear &
 not o,a,c are_collinear & a,b // b1,a1 & a,c // c1,a1 holds
  b,c // c1,b1
  proof
    let o,a,b,c,a1,b1,c1 be Element of OAS such that
A2: a,o // o,a1 and
A3: b,o // o,b1 and
A4: c,o // o,c1 and
A5: not o,a,b are_collinear and
A6: not o,a,c are_collinear and
A7: a,b // b1,a1 and
A8: a,c // c1,a1;
    reconsider y=o,u=a,v=b,w=c,u1=a1,v1=b1,w1=c1 as VECTOR of V by A1,Th3;
A9: o<>a by A5,DIRAF:31;
A10: now
A11:  not y,u '||' y,v & not y,u '||' y,w
      proof
        assume not thesis;
        then y,u // y,v or y,u // v,y or y,u // y,w or y,u // w,y by
GEOMTRAP:def 1;
        then o,a // o,b or o,a // b,o or o,a // o,c or o,a // c,o by A1,
GEOMTRAP:2;
        then o,a '||' o,b or o,a '||' o,c by DIRAF:def 4;
        hence contradiction by A5,A6,DIRAF:def 5;
      end;
      o,c // c1,o by A4,DIRAF:2;
      then y,w // w1,y by A1,GEOMTRAP:2;
      then
A12:  y,w '||' y,w1 by GEOMTRAP:def 1;
      o,b // b1,o by A3,DIRAF:2;
      then y,v // v1,y by A1,GEOMTRAP:2;
      then
A13:  y,v '||' y,v1 by GEOMTRAP:def 1;
      assume
A14:  o<>a1;
      u,y // y,u1 by A1,A2,GEOMTRAP:2;
      then consider r being Real such that
A15:  y-u = r*(u1-y) and
A16:  0<r by A9,A14,Lm1;
      u,w // w1,u1 by A1,A8,GEOMTRAP:2;
      then u,w '||' u1,w1 by GEOMTRAP:def 1;
      then
A17:  w1 = u1 + (-r)"*(w-u) by A15,A16,A12,A11,Th10;
      u,v // v1,u1 by A1,A7,GEOMTRAP:2;
      then u,v '||' u1,v1 by GEOMTRAP:def 1;
      then v1 = u1 + (-r)"*(v-u) by A15,A16,A13,A11,Th10;
      then
A18:  1*(w1-v1) = (u1+(-r)"*(w-u)) - (u1+(-r)"*(v-u)) by A17,RLVECT_1:def 8
        .= (((-r)"*(w-u)+u1) - u1)-(-r)"*(v-u) by RLVECT_1:27
        .= (-r)"*(w-u) - (-r)"*(v-u) by RLSUB_2:61
        .= (-r)"*((w-u) - (v-u)) by RLVECT_1:34
        .= (-r)"*(w - ((v-u)+u)) by RLVECT_1:27
        .= (-r)"*(w - v) by RLSUB_2:61
        .= (-(r)")*(w-v) by XCMPLX_1:222
        .= r"*(-(w-v)) by RLVECT_1:24
        .= r"*(v-w) by RLVECT_1:33;
      0<r" by A16,XREAL_1:122;
      then w,v // v1,w1 by A18,ANALOAF:def 1;
      then c,b // b1,c1 by A1,GEOMTRAP:2;
      hence thesis by DIRAF:2;
    end;
    now
      assume
A19:  o=a1;
A20:  o=c1
      proof
        c,o '||' o,c1 by A4,DIRAF:def 4;
        then o,c1 '||' o,c by DIRAF:22;
        then
A21:    o,c1,c are_collinear by DIRAF:def 5;
A22:    o,c1,o are_collinear by DIRAF:31;
        assume
A23:    o<>c1;
        a,c '||' c1,o by A8,A19,DIRAF:def 4;
        then o,c1 '||' c,a by DIRAF:22;
        then o,c1,a are_collinear by A23,A21,DIRAF:33;
        hence contradiction by A6,A23,A21,A22,DIRAF:32;
      end;
      o=b1
      proof
        b,o '||' o,b1 by A3,DIRAF:def 4;
        then o,b1 '||' o,b by DIRAF:22;
        then
A24:    o,b1,b are_collinear by DIRAF:def 5;
A25:    o,b1,o are_collinear by DIRAF:31;
        assume
A26:    o<>b1;
        a,b '||' b1,o by A7,A19,DIRAF:def 4;
        then o,b1 '||' b,a by DIRAF:22;
        then o,b1,a are_collinear by A26,A24,DIRAF:33;
        hence contradiction by A5,A26,A24,A25,DIRAF:32;
      end;
      hence thesis by A20,DIRAF:4;
    end;
    hence thesis by A10;
  end;
  hence thesis;
end;
