reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;
reserve B,C for Coset of W;

theorem
  u in C & v in C implies ex v1 st v1 in W & u + v1 = v
proof
  assume u in C & v in C;
  then C = u + W & C = v + W by Th78;
  hence thesis by Th66;
end;
