reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;
reserve G for non empty multMagma;
reserve A for non empty set,
  a for Element of A,
  p for FinSequence of A,
  m1,m2 for Multiset of A;

theorem
  |.<*a*>.| = chi a
proof
A1: rng <*a*> = {a} by FINSEQ_1:39;
  now
    let b be Element of A;
    set x = b;
A2: dom <*a*> = Seg 1 & card Seg 1 = 1 by FINSEQ_1:38,57;
    a <> x implies {x} misses {a} by ZFMISC_1:11;
    then
A3: a <> x implies {x}/\{a} = {};
A4: (chi a).a = 1 & {a}|`<*a*> = <*a*> by A1,Th31;
    <*a*> = (rng<*a*>)|`<*a*>;
    then {x}|`<*a*> = ({x}/\rng<*a*>)|`<*a*> by RELAT_1:96;
    then x <> a implies {x}|`<*a*> = {} & (chi a).b = 0 by A1,A3,Th31;
    hence |.<*a*>.|.x = (chi a).x by A2,A4,Def7,CARD_1:27,RELAT_1:38;
  end;
  hence thesis by Th32;
end;
