theorem
  Mid x,y,t & Mid x,z,t implies Mid x,y,z or Mid x,z,y
proof
  assume that
A1: Mid x,y,t and
A2: Mid x,z,t;
A3: x,z // z,t by A2;
A4: x,y // y,t by A1;
  now
    x,z // x,t by A3,ANALOAF:def 5;
    then
A5: x,t // x,z by Th2;
    assume
A6: x<>t;
    x,y // x,t by A4,ANALOAF:def 5;
    then x,y // x,z by A6,A5,Th3;
    hence thesis by Th7;
  end;
  hence thesis by A1,A2,Th8;
end;
