reserve AS for AffinSpace,
  a,b,c,d,p,q,r,s,x for Element of AS;

theorem Th1:
  (ex a,b,c st LIN a,b,c & a<>b & a<>c & b<>c) implies for p,q
  holds ex r st LIN p,q,r & p<>r & q<>r
proof
  given a,b,c such that
A1: LIN a,b,c and
A2: a<>b and
A3: a<>c and
A4: b<>c;
  let p,q;
A5: now
    assume that
A6: LIN a,b,p and
A7: LIN a,b,q;
A8: LIN a,b,b by AFF_1:7;
A9: LIN a,b,a by AFF_1:7;
    now
      assume
A10:  p = c or q = c;
      now
        assume p <> a or q <> b;
A11:    now
          assume that
A12:      p = c and
A13:      p<>a;
          q = b implies thesis by A1,A2,A9,A8,A12,A13,AFF_1:8;
          hence thesis by A1,A2,A4,A7,A8,A12,AFF_1:8;
        end;
        now
          assume that
A14:      q = c and
          q<>b;
          p = b implies thesis by A1,A2,A3,A9,A8,A14,AFF_1:8;
          hence thesis by A1,A2,A4,A6,A8,A14,AFF_1:8;
        end;
        hence thesis by A3,A4,A10,A11;
      end;
      hence thesis by A1,A3,A4;
    end;
    hence thesis by A1,A2,A6,A7,AFF_1:8;
  end;
A15: now
    assume that
A16: not LIN a,b,p and
    not LIN a,b,q;
    now
      consider p9 being Element of AS such that
A17:  a,b // p,p9 and
A18:  p<>p9 by DIRAF:40;
      assume
A19:  not p,q // a,b;
A20:  not LIN p,p9,q
      proof
        assume LIN p,p9,q;
        then p,p9 // p,q by AFF_1:def 1;
        hence contradiction by A19,A17,A18,AFF_1:5;
      end;
      p,p9 // a,b by A17,AFF_1:4;
      then
      ex p99 being Element of AS st LIN p,p9,p99 & p<>p99 & p9<>p99 by A1,A2,A3
,A4,A16,Lm3;
      then consider r such that
A21:  LIN q,p,r and
A22:  q<>r & p<>r by A18,A20,Lm1;
      LIN p,q,r by A21,AFF_1:6;
      hence thesis by A22;
    end;
    hence thesis by A1,A2,A3,A4,A16,Lm3;
  end;
  now
    assume ( not LIN a,b,q)& LIN a,b,p;
    then consider x such that
A23: LIN q,p,x and
A24: q<>x & p<>x by A1,A2,A3,A4,Lm2;
    LIN p,q,x by A23,AFF_1:6;
    hence thesis by A24;
  end;
  hence thesis by A1,A2,A3,A4,A5,A15,Lm2;
end;
