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 Th78:
  u in C iff C = u + W
proof
  thus u in C implies C = u + W
  proof
    assume
A1: u in C;
    ex v st C = v + W by Def6;
    hence thesis by A1,Th54;
  end;
  thus thesis by Th43;
end;
