reserve r,s,t,u for Real;

theorem Th24:
  for X being RealLinearSpace, M,N being Subset of X, v being
  Point of X holds v+M meets N iff v in N+(-M)
proof
  let X be RealLinearSpace, M,N be Subset of X, v being Point of X;
A1: N+(-1)*M = {u + w where u,w is Point of X: u in N & w in (-1)*M} by
RUSUB_4:def 9;
  hereby
A2: v+M = {v + u where u is Point of X: u in M} by RUSUB_4:def 8;
    assume v+M meets N;
    then consider z being object such that
A3: z in v+M and
A4: z in N by XBOOLE_0:3;
    consider u being Point of X such that
A5: v+u = z and
A6: u in M by A3,A2;
    reconsider z as Point of X by A3;
A7: (-1)*u in (-1)*M by A6;
    z + (-1)*u = v + (u + (-1)*u) by A5,RLVECT_1:def 3
      .= v + (u + -u) by RLVECT_1:16
      .= v+ 0.X by RLVECT_1:5
      .= v;
    hence v in N+-M by A4,A7,Th3;
  end;
  assume v in N+(-M);
  then consider u,w being Point of X such that
A8: v = u+w and
A9: u in N and
A10: w in (-1)*M by A1;
  consider w9 being Point of X such that
A11: w = (-1)*w9 and
A12: w9 in M by A10;
A13: (-1)*w = (-1)*(-1)*w9 by A11,RLVECT_1:def 7
    .= w9 by RLVECT_1:def 8;
  v+w9 = u + (w+w9) by A8,RLVECT_1:def 3
    .= u + (w+-w) by A13,RLVECT_1:16
    .= u + 0.X by RLVECT_1:5
    .= u;
  then u in v+M by A12,Lm1;
  hence thesis by A9,XBOOLE_0:3;
end;
