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
  support m c= J implies m|J = m
  proof
    assume
A1: support m c= J;
    let i be object;
    assume
A2: i in I;
    per cases;
    suppose i in J;
      hence thesis by A2,BAR;
    end;
    suppose not i in J;
      then (m|J).i = 0 & not i in support m by A1,A2,BAR;
      hence thesis by PRE_POLY:def 7;
    end;
  end;
