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 Th35:
  for f be Function st z in dom f holds Ext(<*f.z*>,x,y) = <*Ext(f,x,y).z*>
proof
  let f be Function;
  assume
A1: z in dom f;
A2:  1 in Seg 1;
A3:  dom Ext(<*f.z*>,x,y) = dom <*f.z*> = Seg 1 by Def5,FINSEQ_1:38;
  then
A4: len Ext(<*f.z*>,x,y) = len <*f.z*> =1 by FINSEQ_3:29,FINSEQ_1:40;
A5:<*f.z*>.1 = f.z;
  per cases;
  suppose x in f.z;
    then Ext(<*f.z*>,x,y).1 = (f.z)\/{y} = Ext(f,x,y).z by A2,Def5,A3,A5,A1;
    hence thesis by A4,FINSEQ_1:40;
  end;
  suppose not x in f.z;
    then Ext(<*f.z*>,x,y).1 = (f.z) = Ext(f,x,y).z by A2,Def5,A3,A5,A1;
    hence thesis by A4,FINSEQ_1:40;
  end;
end;
