reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;
reserve i,j,k for Element of NAT;
reserve a,b,c,d for Real;
reserve y,r,s,x,t,w for Element of RAT+;

theorem Th5:
  for x1,x2,x3,x4,y1,y2,y3,y4 being Real st [*x1,x2,x3,x4*]
  = [*y1,y2,y3,y4*] holds x1 = y1 & x2 = y2 & x3=y3 & x4=y4
proof
  let x1,x2,x3,x4,y1,y2,y3,y4 be Real such that
A1: [*x1,x2,x3,x4*] = [*y1,y2,y3,y4*];
   reconsider xx1=x1, xx2=x2, yy1=y1, yy2=y2 as Element of REAL
            by XREAL_0:def 1;
  per cases;
  suppose
A2: x3=0 & x4=0;
    then
A3: [*x1,x2,x3,x4*] = [*xx1,xx2*] by Lm3;
A4: now
      assume y3<>0 or y4<>0;
      then [*xx1,xx2*] = (0,1,2,3) --> (y1,y2,y3,y4) by A1,A3,Def5;
      hence contradiction by Th4;
    end;
    then [*y1,y2,y3,y4*] = [*yy1,yy2*] by Lm3;
    hence thesis by A1,A2,A3,A4,ARYTM_0:10;
  end;
  suppose x3<>0 or x4<>0;
    then
A5: [*y1,y2,y3,y4*] = (0,1,2,3) --> (x1,x2,x3,x4) by A1,Def5;
    now
      assume that
A6:   y3=0 and
A7:   y4=0;
      [*y1,y2,y3,y4*] = [*yy1,yy2*] by A6,A7,Lm3;
      hence contradiction by A5,Th4;
    end;
    then [*y1,y2,y3,y4*] = (0,1,2,3) --> (y1,y2,y3,y4) by Def5;
    hence thesis by A5,Lm1,FUNCT_4:146;
  end;
end;
