reserve
  a for natural Number,
  k,l,m,n,k1,b,c,i for Nat,
  x,y,z,y1,y2 for object,
  X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for FinSequence;
reserve D for set;
reserve a, b, c, d, e, f for object;
reserve x1,x2,y1,y2 for object;

theorem
  <*x1,x2*> = {[1,x1], [2,x2]}
proof
  reconsider f={[1,x1], [2,x2]} as Function by GRFUNC_1:8;
A1: dom f = {1,2} by RELAT_1:10;
  then
A2: dom <*x1,x2*> = dom f by Th2,Th88;
  now
    let x be object;
    assume
A3: x in {1,2};
    per cases by A3,TARSKI:def 2;
    suppose
A4:   x = 1;
      then
A5:   <*x1,x2*>.x = x1;
      [x,x1] in f by A4,TARSKI:def 2;
      hence f.x = <*x1,x2*>.x by A5,FUNCT_1:1;
    end;
    suppose
A6:   x = 2;
      then
A7:   <*x1,x2*>.x = x2;
      [x,x2] in f by A6,TARSKI:def 2;
      hence f.x = <*x1,x2*>.x by A7,FUNCT_1:1;
    end;
  end;
  hence thesis by A1,A2,FUNCT_1:2;
end;
