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
  Mid a,b,c & Mid a,c,d implies Mid b,c,d
proof
  assume that
A1: Mid a,b,c and
A2: Mid a,c,d;
  now
A3: a,b // b,c by A1;
    then a,b // a,c by ANALOAF:def 5;
    then
A4: b,c // a,c or a=b by A3,ANALOAF:def 5;
    assume
A5: a<>c;
    a,c // c,d by A2;
    then b,c // c,d by A5,A4,Th3;
    hence thesis;
  end;
  hence thesis by A1,A2,Th8;
end;
