reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem Th103:
  qtrap o,p implies o<>p
proof
  ex b st o<>b
  proof
    consider x,y,z such that
A1: x<>y and
    y<>z and
    z<>x by Th102;
    o<>x or o<>y or o<>z by A1;
    hence thesis;
  end;
  then consider b such that
A2: o<>b;
  consider c such that
A3: not o,b // o,c by A2,Th13;
  assume qtrap o,p;
  then consider d such that
A4: o,p,b are_collinear implies o,c,d are_collinear & p,c // b,d;
A5: now
    assume that
A6: b<>d & not o,b // o,c and
A7: o,d // b,d and
A8: o,c // b,d;
    d,o // d,b by A7,Th6;
    hence b<>d & not o,b // o,c & b,d // o,b & b,d // o,c by A6,A8,Th6,Th7;
  end;
  assume not thesis;
  then o,o // o,b implies o,c // o,d & o,c // b,d by A4;
  then
  b=d & not o,b // o,c & o,c // o,d or b<>d & o<>c & not o,b // o,c & o,c
  // o,d & o,c // b,d by A3,Def1,Th3;
  hence contradiction by A5,Def1,Th6;
end;
