reserve x,y for set;
reserve X for non empty set;
reserve a,b,c,d for Element of X;
reserve S for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u,w for Element of S;

theorem Th27:
  z,x '||' x,t & x<>z implies ex u st y,x '||' x,u & y,z '||' t,u
proof
  assume that
A1: z,x '||' x,t and
A2: x<>z;
A3: now
    consider p such that
A4: Mid z,x,p and
A5: x<>p by Th13;
A6: z,x // x,p by A4;
    then consider q such that
A7: y,x // x,q and
A8: y,z // p,q by A2,ANALOAF:def 5;
    assume
A9: z,x // t,x;
    then x,p // t,x by A2,A6,ANALOAF:def 5;
    then p,x // x,t by Th2;
    then consider r such that
A10: q,x // x,r and
A11: q,p // t,r by A5,ANALOAF:def 5;
A12: now
      assume q=p;
      then
A13:  x,p // y,x by A7,Th2;
      x,p // z,x by A6,Th2;
      then z,x // y,x by A5,A13,ANALOAF:def 5;
      then y,x // t,x by A2,A9,ANALOAF:def 5;
      then
A14:  y,x '||' x,t;
      y,z // t,t by ANALOAF:def 5;
      then y,z '||' t,t;
      hence thesis by A14;
    end;
A15: now
A16:  x,z // x,t by A9,Th2;
      assume
A17:  q=x;
      p,x // x,z by A6,Th2;
      then y,z // x,z by A5,A8,A17,Th3;
      then y,z // x,t by A2,A16,Th3;
      then
A18:  y,z '||' t,x;
      y,x // x,x by Th4;
      then y,x '||' x,x;
      hence thesis by A18;
    end;
    now
      assume that
A19:  q<>p and
A20:  q<>x;
      x,q // r,x by A10,Th2;
      then y,x // r,x by A7,A20,Th3;
      then
A21:  y,x '||' x,r;
      p,q // r,t by A11,Th2;
      then y,z // r,t by A8,A19,Th3;
      then y,z '||' t,r;
      hence thesis by A21;
    end;
    hence thesis by A12,A15;
  end;
  now
    assume z,x // x,t;
    then consider u such that
A22: y,x // x,u & y,z // t,u by A2,ANALOAF:def 5;
    y,x '||' x,u & y,z '||' t,u by A22;
    hence thesis;
  end;
  hence thesis by A1,A3;
end;
