reserve a,b for object, I,J for set;
reserve b for bag of I;
reserve R for asymmetric transitive non empty RelStr,
  a,b,c for bag of the carrier of R,
  x,y,z for Element of R;
reserve p for partition of b-'a, q for partition of b;
reserve J for set, m for bag of I;

theorem Th27A:
  support(m-'(m|J)) = (support m)\J
  proof
    thus support(m-'(m|J)) c= (support m)\J
    proof let x be object;
      assume x in support(m-'(m|J));
      then (m-'(m|J)).x <> 0 by PRE_POLY:def 7;
      then m.x-'(m|J).x <> 0 by PRE_POLY:def 6;
      then m.x-'(m|J).x > 0;
      then m.x-(m|J).x > 0 by XREAL_0:def 2;
      then m.x > (m|J).x by XREAL_1:47;
      then m.x <> (m|J).x & x in dom m = I & m.x > 0
      by PARTFUN1:def 2,FUNCT_1:def 2;
      then not x in J & x in support m by BAR,PRE_POLY:def 7;
      hence x in (support m)\J by XBOOLE_0:def 5;
    end;
    let x be object;
    assume x in (support m)\J;
    then
A1: x in (support m) & not x in J by XBOOLE_0:def 5; then
A2: m.x <> 0 by PRE_POLY:def 7;
    then x in dom m = I by PARTFUN1:def 2,FUNCT_1:def 2;
    then (m|J).x = 0 by A1,BAR;
    then (m-'(m|J)).x = m.x-'(m|J).x <> 0 by A2,NAT_D:40,PRE_POLY:def 6;
    hence thesis by PRE_POLY:def 7;
  end;
