
theorem Th27:
  for I be non empty set,
      F be Group-Family of I,
      i be Element of I,
      a,b be Function
  st dom a = I & b = a +* (i,1_F.i)
  holds support(b,F) = support(a,F) \ {i}
  proof
    let I be non empty set,
        F be Group-Family of I,
        i be Element of I,
        a,b be Function;
    assume that
    A2: dom a = I and
    A3: b = a +* (i,1_F.i);
    for j be object holds j in support(b,F) iff j in support(a,F) \ {i}
    proof
      let j be object;
      hereby
        assume j in support(b,F); then
        consider Z being Group such that
A:      Z = F.j & b.j <> 1_Z & j in I by Def1;
        A8: j <> i by A,A2,A3,FUNCT_7:31; then
        {j} misses {i} by ZFMISC_1:11; then
        A9: not j in {i} by ZFMISC_1:48;
        a.j = b.j by A3,A8,FUNCT_7:32; then
        j in support(a,F) by A,Def1;
        hence j in support(a,F) \ {i} by A9,XBOOLE_0:def 5;
      end;
      assume j in support(a,F) \ {i}; then
      A10: j in support(a,F) & not j in {i} by XBOOLE_0:def 5;
      then consider Z being Group such that
A:    Z = F.j & a.j <> 1_Z & j in I by Def1;
      {j} misses {i} by A10,ZFMISC_1:50; then
      b.j = a.j by A3,FUNCT_7:32;
      hence j in support(b,F) by A,Def1;
    end;
    hence thesis by TARSKI:2;
  end;
