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
  b<>c & Mid a,b,c & Mid b,c,d implies Mid a,c,d
proof
  assume that
A1: b<>c and
A2: Mid a,b,c and
A3: Mid b,c,d;
  now
    assume
A4: a<>b;
A5: a,b // b,c by A2;
    b,c // c,d by A3;
    then
A6: a,b // c,d by A1,A5,Th3;
    a,b // a,c by A5,ANALOAF:def 5;
    then a,c // c,d by A4,A6,ANALOAF:def 5;
    hence thesis;
  end;
  hence thesis by A3;
end;
