reserve x,y for Element of REAL;
reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem
  for x1,x2,y1,y2 being Element of REAL st [*x1,x2*] = [*y1,y2*] holds
  x1 = y1 & x2 = y2
proof
  let x1,x2,y1,y2 be Element of REAL such that
A1: [*x1,x2*] = [*y1,y2*];
  per cases;
  suppose
A2: x2 = 0;
    then
A3: [*x1,x2*] = x1 by Def5;
A4: now
      assume y2 <> 0;
      then x1 = (0,1) --> (y1,y2) by A1,A3,Def5;
      hence contradiction by Th8;
    end;
    hence x1 = y1 by A1,A3,Def5;
    thus thesis by A2,A4;
  end;
  suppose
    x2 <> 0;
    then
A5: [*y1,y2*] = (0,1) --> (x1,x2) by A1,Def5;
    now
      assume y2 = 0;
      then [*y1,y2*] = y1 by Def5;
      hence contradiction by A5,Th8;
    end;
    then [*y1,y2*] = (0,1) --> (y1,y2) by Def5;
    hence thesis by A5,FUNCT_4:68;
  end;
end;
