
theorem Th28:
  for I be non empty set,
      G be Group,
      i be object,
      a,b be Function of I,G
  st i in support(a) & b = a +* (i,1_G)
  holds support(b) = support(a) \ {i}
  proof
    let I be non empty set,
        G be Group,
        i be object,
        a,b be Function of I,G;
    assume that
    A1: i in support(a) and
    A2: b = a +* (i,1_G);
    A4: dom a = I by FUNCT_2:def 1;
    A5: b.i = 1_G by A1,A2,A4,FUNCT_7:31;
    for j be object holds j in support(b) iff j in support(a) \ {i}
    proof
      let j be object;
      hereby
        assume j in support(b); then
        A7: b.j <> 1_G & j in I by Def2; then
        {j} misses {i} by A5,ZFMISC_1:11; then
        A9: not j in {i} by ZFMISC_1:48;
        a.j = b.j by A2,A5,A7,FUNCT_7:32; then
        j in support(a) by A7,Def2;
        hence j in support(a) \ {i} by A9,XBOOLE_0:def 5;
      end;
      assume j in support(a) \ {i}; then
      A10: j in support(a) & not j in {i} by XBOOLE_0:def 5;
      {j} misses {i} by A10,ZFMISC_1:50; then
      A11: b.j = a.j by A2,FUNCT_7:32;
      a.j <> 1_G & j in I by A10,Def2;
      hence j in support(b) by A11,Def2;
    end;
    hence thesis by TARSKI:2;
  end;
