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 Th74:
  v + W = (- v) + W iff v in W
proof
  thus v + W = (- v) + W implies v in W
  proof
    assume v + W = (- v) + W;
    then v in (- v) + W by Th62;
    then consider u such that
A1: v = - v + u and
A2: u in W;
    0.V = v - (- v + u) by A1,RLVECT_1:15
      .= (v - (- v)) - u by RLVECT_1:27
      .= (v + v) - u by RLVECT_1:17
      .= (1r * v + v) - u by Def5
      .= (1r * v + 1r * v) - u by Def5
      .= (1r+1r) * v - u by Def3;
    then (1r+1r)" * ((1r+1r) * v) = (1r+1r)" * u by RLVECT_1:21;
    then ((1r+1r)" * (1r+1r)) * v = (1r+1r)" * u by Def4;
    then v = (1r+1r)" * u by Def5;
    hence thesis by A2,Th40;
  end;
  assume
A3: v in W;
  then v + W = the carrier of W by Lm6;
  hence thesis by A3,Th70;
end;
