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 Th84:
  (ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W
proof
  thus (ex v st v1 in v + W & v2 in v + W) implies v1 - v2 in W
  proof
    given v such that
A1: v1 in v + W and
A2: v2 in v + W;
    consider u2 such that
A3: u2 in W and
A4: v2 = v + u2 by A2,Th82;
    consider u1 such that
A5: u1 in W and
A6: v1 = v + u1 by A1,Th82;
    v1 - v2 = (u1 + v) + ((- v) - u2) by A6,A4,RLVECT_1:30
      .= ((u1 + v) + (- v)) - u2 by RLVECT_1:def 3
      .= (u1 + (v + (- v))) - u2 by RLVECT_1:def 3
      .= (u1 + 0.V) - u2 by RLVECT_1:5
      .= u1 - u2 by RLVECT_1:4;
    hence thesis by A5,A3,Th42;
  end;
  assume v1 - v2 in W;
  then
A7: - (v1 - v2) in W by Th41;
  take v1;
  thus v1 in v1 + W by Th62;
  v1 + (- (v1 - v2)) = v1 + ((- v1) + v2) by RLVECT_1:33
    .= (v1 + (- v1)) + v2 by RLVECT_1:def 3
    .= 0.V + v2 by RLVECT_1:5
    .= v2 by RLVECT_1:4;
  hence thesis by A7;
end;
