reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th60:
  i in dom f implies (App <*f*>).<*i*> = <*f.i*>
proof
  set I=<*i*>, A=App <*f*>;
A1: len I = 1 & I.1 =i by FINSEQ_1:40;
  assume i in dom f;
  then
A2: I in doms <*f*> by A1,Th51;
A3: 1 in dom I by A1,FINSEQ_3:25;
  <*f*>.1 .(I.1) = f.i;
  then (A.I).1 = f.i by A3,A2,Def9;
  hence thesis by A2,A1,Def9,FINSEQ_1:40;
end;
