reserve F for Field;
reserve S for SymSp of F;
reserve a,b,c,d,a9,b9,p,q,r,s,x,y,z for Element of S;
reserve k,l for Element of F;

theorem Th11:
  a-b _|_ d & a-c _|_ d implies b-c _|_ d
proof
  assume that
A1: a-b _|_ d and
A2: a-c _|_ d;
  -(a-b) _|_ d by A1,Th6;
  then -a+b _|_ d by VECTSP_1:17;
  then b+(-a)+(a-c) _|_ d by A2,Def1;
  then b+((-a)+(a-c)) _|_ d by RLVECT_1:def 3;
  then b+((-a+a)+(-c)) _|_ d by RLVECT_1:def 3;
  then b+((0.S)+(-c)) _|_ d by RLVECT_1:5;
  hence thesis by RLVECT_1:4;
end;
