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 Th44:
  support (m|J) = J /\ support m
  proof set f = m|J;
    thus support f c= J /\ support m
    proof let x be object;
      assume x in support f;
      then
A3:   f.x <> 0 by PRE_POLY:def 7;
      then
A4:   x in dom f = I by PARTFUN1:def 2,FUNCT_1:def 2;
      then
A5:   x in J by BAR,A3;
      then f.x = m.x by BAR,A4;
      then x in support m by A3,PRE_POLY:def 7;
      hence thesis by A5,XBOOLE_0:def 4;
    end;
    let x be object;
    assume x in J /\ support m;
    then
A1: x in J & x in support m by XBOOLE_0:def 4;
    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 f.x = m.x by A1,BAR;
    hence thesis by A2,PRE_POLY:def 7;
  end;
