reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th130:
  doms <*f*> = <*dom f*> & rngs <*f*> = <*rng f*>
proof
A1: dom doms <*f*> = dom <*f*> & dom <*dom f*> = Seg 1
      by FINSEQ_1:38,FUNCT_6:def 2;
A2: <*f*>.1 = f;
A3: dom <*f*> = Seg 1 & {f} = {f} by FINSEQ_1:38;
A5: now
    let x be object;
    assume
A6: x in {1};
    then x = 1 by TARSKI:def 1;
    hence (rngs <*f*>).x = <*rng f*>.x
     by A2,A3,A6,FINSEQ_1:2,FUNCT_6:def 3;
  end;
  now
    let x be object;
    assume
A8: x in {1};
    then x = 1 by TARSKI:def 1;
    hence (doms <*f*>).x = <*dom f*>.x
       by A2,A3,A8,FINSEQ_1:2,FUNCT_6:def 2;
  end;
  hence doms <*f*> = <*dom f*> by A1,A3,FINSEQ_1:2;
  dom rngs <*f*> = dom <*f*> & dom <*rng f*> = Seg 1
   by FINSEQ_1:38,FUNCT_6:def 3;
  hence thesis by A3,A5,FINSEQ_1:2;
end;
