reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
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 [:COMPLEX,D:],D;
reserve B,C for Coset of W;

theorem Th85:
  v + W = u + W implies ex v1 st v1 in W & v + v1 = u
proof
  assume v + W = u + W;
  then v in u + W by Th62;
  then consider u1 such that
A1: v = u + u1 and
A2: u1 in W;
  take v1 = u - v;
  0.V = (u + u1) - v by A1,RLVECT_1:15
    .= u1 + (u - v) by RLVECT_1:def 3;
  then v1 = - u1 by RLVECT_1:def 10;
  hence v1 in W by A2,Th41;
  thus v + v1 = (u + v) - v by RLVECT_1:def 3
    .= u + (v - v) by RLVECT_1:def 3
    .= u + 0.V by RLVECT_1:15
    .= u by RLVECT_1:4;
end;
