reserve A,B,C for Ordinal,
        o for object,
        x,y,z,t,r,l for Surreal,
        X,Y for set;

theorem Th51:
  (for x,y holds x*y is Surreal) &
  (for x,y holds x*y = y*x)&
  (for x1,x2,y,x1y,x2y be Surreal st x1 == x2 & x1y = x1*y & x2y = x2*y
     holds  x1y == x2y) &
  (for x1,x2,y1,y2,x1y2,x2y1,x1y1,x2y2 be Surreal st
        x1y1=x1*y1 & x1y2=x1*y2 & x2y1=x2*y1 & x2y2=x2*y2 & x1 < x2 & y1 < y2
     holds x1y2+x2y1 < x1y1+x2y2)
proof
  defpred P0[Ordinal,Surreal,Surreal] means
  born $2 (+) born $3 c= $1 implies $2*$3 = $3 * $2;
  defpred P1[Ordinal,Surreal,Surreal] means
  born $2 (+) born $3 c= $1 implies $2*$3 is Surreal;
  defpred P2[Ordinal,Surreal,Surreal,Surreal] means
      for x1y,x2y be Surreal st
      born $2 (+) born $4 c= $1 & born $3 (+) born $4 c= $1 &
          $2 == $3 & x1y = $2*$4 & x2y = $3*$4
    holds  x1y == x2y;
  defpred P3[Ordinal,Surreal,Surreal,Surreal,Surreal] means
    for x1y2,x2y1,x1y1,x2y2 be Surreal st
        born $2 (+) born $4 c= $1 & born $3 (+) born $4 c= $1 &
        born $2 (+) born $5 c= $1 & born $3 (+) born $5 c= $1 &
        x1y1=$2*$4 & x1y2=$2*$5 & x2y1=$3*$4 & x2y2=$3*$5 & $2 < $3 & $4 < $5
    holds
       x1y2+x2y1 < x1y1+x2y2;
  defpred P4[Ordinal,Surreal,Surreal,Surreal] means
           $3 < $4 implies (for x st x in L_$2 holds P3[$1,x,$2,$3,$4])&
                           (for x st x in R_$2 holds P3[$1,$2,x,$3,$4]);
  defpred Q0[Ordinal] means for x,y holds P0[$1,x,y];
  defpred Q1[Ordinal] means for x,y holds P1[$1,x,y];
  defpred Q2[Ordinal] means for x1,x2,y be Surreal holds P2[$1,x1,x2,y];
  defpred Q3[Ordinal] means for x1,x2,y1,y2 be Surreal
    holds P3[$1,x1,x2,y1,y2];
  defpred Q[Ordinal] means Q0[$1] & Q1[$1] & Q2[$1] & Q3[$1];
  A1: for D be Ordinal st for C be Ordinal st C in D holds Q[C] holds Q[D]
  proof
    let D be Ordinal such that
    A2: for C be Ordinal st C in D holds Q[C];
    thus A3: for x,y holds P0[D,x,y]
    proof
      A4: for x,y st born x (+) born y c= D
      for X,Y be surreal-membered set st (X c= L_x\/R_x & Y c= L_y\/R_y)
      holds comp(X,x,y,Y) c= comp(Y,y,x,X)
      proof
        let x,y be Surreal such that A5: born x (+) born y c= D;
        let X,Y be surreal-membered set such that
        A6:X c= L_x\/R_x  & Y c= L_y\/R_y;
        let a be object;
        assume a in comp(X,x,y,Y);
        then consider x1,y1 be Surreal such that
        A7: a = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by Def14;
        A8: born x1 (+) born y1 in born x1 (+) born y &
        born x (+) born y1 in born x (+) born y &
        born x1 (+) born y in born x (+) born y
        by SURREALO:1,A6,A7,ORDINAL7:94;
        then born x1 (+) born y1 in born x (+) born y by ORDINAL1:10;
        then A9:x1*y1 = y1*x1 & x*y1 = y1*x & x1*y = y*x1 by A8,A2,A5;
        reconsider yx1=y*x1,y1x=y1*x as Surreal by A8,A2,A5;
        (y*x1) +' (y1*x) =y1x +yx1 by Def11
        .=(y1*x) +' (y*x1);
        hence thesis by A7,Def14,A9;
      end;
      A10:for x,y st born x (+) born y c= D
      for X,Y be surreal-membered set st (X c= L_x\/R_x & Y c= L_y\/R_y)
      holds comp(X,x,y,Y) = comp(Y,y,x,X)
      proof
        let x,y such that A11: born x (+) born y c= D;
        let X,Y be surreal-membered set such that
        A12: X c= L_x\/R_x & Y c= L_y\/R_y;
        comp(X,x,y,Y) c= comp(Y,y,x,X) c= comp(X,x,y,Y)by A11,A12, A4;
        hence thesis by XBOOLE_0:def 10;
      end;
      let x,y be Surreal such that A13: born x (+) born y c= D;
      L_x c= L_x \/ R_x & R_x c= L_x \/ R_x &
      L_y c= L_y\/R_y & R_y c= L_y\/R_y by XBOOLE_1:7;
      then A14: comp(L_x,x,y,L_y) = comp(L_y,y,x,L_x) &
      comp(R_x,x,y,R_y) = comp(R_y,y,x,R_x) &
      comp(L_x,x,y,R_y) = comp(R_y,y,x,L_x) &
      comp(R_x,x,y,L_y) = comp(L_y,y,x,R_x) by A10,A13;
      x*y = [comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y),
      comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y)] by Th50
      .=y*x by A14,Th50;
      hence thesis;
    end;
    thus for x,y holds P1[D,x,y]
    proof
      let x,y;
      assume A15:born x (+) born y c= D;
      set CC =(comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y)) \/
      (comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y));
      A16:for X,Y be surreal-membered set st X c= L_x\/R_x & Y c=L_y\/R_y
      holds comp(X,x,y,Y) is surreal-membered
      proof
        let X,Y be surreal-membered set such that
        A17: X c= L_x\/R_x & Y c=L_y\/R_y;
        let z be object;
        assume z in comp(X,x,y,Y);
        then consider x1,y1 be Surreal such that
        A18:z = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by Def14;
        A19: born x1 (+) born y1 in born x1 (+) born y &
        born x (+) born y1 in born x (+) born y &
        born x1 (+) born y in born x (+) born y
        by A17,SURREALO:1,A18,ORDINAL7:94;
        then born x1 (+) born y1 in born x (+) born y by ORDINAL1:10;
        then reconsider x1y=x1*y,xy1=x*y1,x1y1=x1*y1 as Surreal by A19,A15,A2;
        thus thesis by A18;
      end;
      A20:L_x c= L_x\/R_x & R_x c= L_x\/R_x &
      L_y c= L_y\/R_y & R_y c= L_y\/R_y by XBOOLE_1:7;
      defpred P[object,object] means $1 is Surreal &
      for z be Surreal st z = $1 holds $2 = born z;
      A21: for x,y,z being object st P[x,y] & P[x,z] holds y = z
      proof
        let x,y,z be object such that A22: P[x,y] & P[x,z];
        reconsider x as Surreal by A22;
        thus y=born x by A22
        .=z by A22;
      end;
      consider OO be set such that
      A23: for z be object holds z in OO iff ex y be object st y in CC & P[y,z]
      from TARSKI_0:sch 1(A21);
      for x be set st x in OO holds x is ordinal
      proof
        let x be set;
        assume x in OO;
        then consider y be object such that
        A24: y in CC & P[y,x] by A23;
        reconsider y as Surreal by A24;
        x= born y by A24;
        hence thesis;
      end;
      then OO is ordinal-membered by ORDINAL6:1;
      then reconsider U=union OO as Ordinal;
      A25: for o be object st o in CC
      ex O be Ordinal st O in succ U & o in Day O
      proof
        let o be object such that A26: o in CC;
        A27: comp(L_x,x,y,L_y) is surreal-membered &
        comp(R_x,x,y,R_y) is surreal-membered &
        comp(L_x,x,y,R_y) is surreal-membered &
        comp(R_x,x,y,L_y) is surreal-membered by A16,A20;
        o in (comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y))
        or o in comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y) by A26,XBOOLE_0:def 3;
        then o in comp(L_x,x,y,L_y) or o in comp(R_x,x,y,R_y)
        or o in comp(L_x,x,y,R_y) or o in comp(R_x,x,y,L_y) by XBOOLE_0:def 3;
        then reconsider o as Surreal by A27;
        P[o,born o];
        then born o c= U by A23,A26,ZFMISC_1:74;
        then A28: born o in succ U by  ORDINAL1:6,12;
        o in Day born o by SURREAL0:def 18;
        hence thesis by A28;
      end;
      comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y) <<
      comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y)
      proof
        let l,r be Surreal;
        assume A29: l in comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y) &
        r in comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y);
        per cases by A29,XBOOLE_0:def 3;
        suppose A30: l in comp(L_x,x,y,L_y) & r in comp(L_x,x,y,R_y);
          then consider xL1,yL be Surreal such that
          A31:  l = (xL1*y) +' (x*yL) +' -' (xL1*yL) & xL1 in L_x & yL in L_y
          by Def14;
          consider xL2,yR be Surreal such that
          A32:  r = (xL2*y) +' (x*yR) +' -' (xL2*yR) & xL2 in L_x & yR in R_y
          by Def14,A30;
          xL1 in L_x \/ R_x & xL2 in L_x \/ R_x by A31,A32,XBOOLE_0:def 3;
          then A33:born xL1 (+) born y in born x (+)born y &
          born xL2 (+) born y in born x (+)born y by SURREALO:1,ORDINAL7:94;
          then reconsider xL1y=xL1*y,xL2y=xL2*y as Surreal by A15,A2;
          set BL = (born xL1 (+) born y) \/ (born xL2 (+) born y);
          A34: BL in born x (+)born y by A33, ORDINAL3:12;
          A35: born xL1 (+) born y c= BL & born xL2 (+) born y c= BL
          by XBOOLE_1:7;
          A36:xL1 == xL2 implies xL1y == xL2y by A35,A34,A15,A2;
          A37:yR in L_y \/ R_y & yL in L_y \/ R_y by A31,A32,XBOOLE_0:def 3;
          then A38:born xL1 (+) born yL in born xL1 (+)born y &
          born xL1 (+) born yR in born xL1 (+)born y &
          born xL2 (+) born yL in born xL2 (+)born y &
          born xL2 (+) born yR in born xL2 (+)born y
          by SURREALO:1,ORDINAL7:94;
          then born xL1 (+) born yL in born x (+)born y &
          born xL1 (+) born yR in born x (+)born y &
          born xL2 (+) born yR in born x (+)born y &
          born xL2 (+) born yL in born x (+)born y
          by A33,ORDINAL1:10;
          then reconsider xL1yR=xL1*yR,xL2yR=xL2*yR,xL2yL=xL2*yL,
          xL1yL=xL1*yL as Surreal by A15,A2;
          set BY=(born x (+) born yL)\/(born x (+) born yR);
          A39: born x (+) born yL in born x (+)born y &
          born x (+) born yR in born x (+)born y by
          A37,SURREALO:1,ORDINAL7:94;
          then reconsider xyL=x*yL,xyR=x*yR as Surreal by A15,A2;
          A40: born xL1 (+) born yR c= BL & born xL2 (+) born yR c= BL &
          born xL2 (+) born yL c= BL &
          born xL1 (+) born yL c= BL by A38,A35,ORDINAL1:def 2;
          A41:xL1 == xL2 implies xL1yR == xL2yR by A40,A34,A15,A2;
          L_y << R_y by SURREAL0:45;
          then A42: yL < yR by A31,A32;
          x <=x & y <= y;
          then A43: L_x << {x} << R_x & x in {x} &
          L_y << {y} << R_y & y in {y} by SURREAL0:43,TARSKI:def 1;
          then A44: xL1<x & xL2<x & yL < y by A31,A32;
          A45: BY in born x (+)born y by A39,ORDINAL3:12;
          A46: born x (+) born yL c= BY & born x (+) born yR c= BY
          by XBOOLE_1:7;
          set BB = BL \/BY;
          A47: BB in born x (+)born y by A34,A45,ORDINAL3:12;
          A48:  BL c= BB & BY c= BB by XBOOLE_1:7;
          then A49: born xL1 (+) born yL c= BB&
          born xL1 (+) born yR c= BB &
          born x (+) born yL c= BB & born x (+) born yR c= BB
          by A40,A46,XBOOLE_1:1;
          A50: born xL1 (+) born y c= BB & born xL2 (+) born y c= BB
          by A35,A48,XBOOLE_1:1;
          A51: born yL (+) born xL2 c= BB & born yR (+) born xL2 c= BB
          by A40,A48,XBOOLE_1:1;
          xL1yR+xyL < xL1yL+xyR by A47,A15,A2,A49,A42,A44;
          then xyL < xL1yL+xyR - xL1yR by Th42;
          then xyL < xL1yL+(xyR - xL1yR) by Th37;
          then A52: xyL - xL1yL < xyR - xL1yR by Th41;
          per cases;
          suppose A53: xL1 < xL2;
            xL2yL + xL1y < xL1yL + xL2y by A53,A44,A49,A50,A51,A47,A15,A2;
            then xL1y < xL1yL + xL2y - xL2yL by Th42;
            then xL1y < xL1yL +(xL2y  - xL2yL) by Th37;
            then xL1y - xL1yL <  xL2y - xL2yL by Th41;
            then A54: xL1y +- xL1yL + xyL <= xL2y - xL2yL + xyL by Th32;
            A55: xL2y - xL2yL + xyL = xyL - xL2yL + xL2y by Th37;
            xL2yR + xyL < xL2yL + xyR by A42,A44,A47,A15,A2,A49,A51;
            then xyL < xL2yL + xyR - xL2yR by Th42;
            then xyL < xL2yL + (xyR - xL2yR) by Th37;
            then xyL - xL2yL < xyR - xL2yR by Th41;
            then xyL - xL2yL + xL2y < xyR - xL2yR +xL2y by Th32;
            then xL1y - xL1yL + xyL < xyR - xL2yR +xL2y
            by SURREALO:4,A54,A55;
            then xL1y +(- xL1yL + xyL) < xyR +- xL2yR +xL2y by Th37;
            then xL1y +(xyL - xL1yL) < xyR +(xL2y- xL2yR) by Th37;
            then xL1y +xyL - xL1yL < xyR +(xL2y- xL2yR) by Th37;
            then (xL1y + xyL) + - (xL1yL) < (xL2y + xyR) + - xL2yR by Th37;
            hence l < r by A31,A32;
          end;
          suppose A56: xL2 < xL1;
            A57: born y (+) born xL1 c= BB & born yR (+) born xL1 c= BB
            by A48,A40,XBOOLE_1:1,A35;
            y < yR & xL2 < xL1 by A56,A43,A32;
            then xL1y + xL2yR < xL2y + xL1yR by A50,A51,A57,A47,A15,A2;
            then xL1y  < xL2y + xL1yR -xL2yR by Th42;
            then xL1y  < xL1yR+ (xL2y  -xL2yR) by Th37;
            then xL1y - xL1yR  <  xL2y  -xL2yR by Th41;
            then xL1y - xL1yR  +xyR <  xL2y  +-xL2yR +xyR by Th32;
            then xL1y - xL1yR  +xyR <  xL2y  +xyR+ -xL2yR by Th37;
            then A58:  xL1y + (xyR+- xL1yR) <  xL2y  +xyR+ -xL2yR by Th37;
            xL1yR + xyL < xL1yL + xyR by A42,A44,A47,A15,A2,A49;
            then xyL < xL1yL + xyR -xL1yR by Th42;
            then xyL < xL1yL + (xyR -xL1yR) by Th37;
            then xyL - xL1yL < xyR -xL1yR by Th41;
            then  xL1y +(xyL - xL1yL) <= xL1y +(xyR -xL1yR) by Th32;
            then xL1y +(xyL - xL1yL) < xL2y  +xyR-xL2yR by A58,SURREALO:4;
            then (xL1y + xyL) + - (xL1yL) < (xL2y + xyR) - xL2yR by Th37;
            hence l < r by A31,A32;
          end;
          suppose A59: xL1 == xL2;
            then - xL1yR <= - xL2yR by A41,Th10;
            then A60: (xL2y + xyR)- xL1yR <= (xL2y + xyR)- xL2yR by Th32;
            xL1y + (xyL - xL1yL) < xL2y +(xyR - xL1yR) by A59,A36,A52,Th44;
            then xL1y + (xyL - xL1yL) < (xL2y +xyR) - xL1yR by Th37;
            then xL1y + (xyL - xL1yL) < (xL2y + xyR)- xL2yR
            by A60,SURREALO:4;
            then (xL1y + xyL) - (xL1yL) < (xL2y + xyR) - xL2yR by Th37;
            hence l < r by A31,A32;
          end;
        end;
        suppose A61: l in comp(R_x,x,y,R_y) & r in comp(R_x,x,y,L_y);
          then consider xR1,yR be Surreal such that
          A62:  l = (xR1*y) +' (x*yR) +' -' (xR1*yR) & xR1 in R_x & yR in R_y
          by Def14;
          consider xR2,yL be Surreal such that
          A63:  r = (xR2*y) +' (x*yL) +' -' (xR2*yL) & xR2 in R_x & yL in L_y
          by Def14,A61;
          xR1 in L_x \/ R_x & xR2 in L_x \/ R_x by A62,A63,XBOOLE_0:def 3;
          then A64:born xR1 (+) born y in born x (+)born y &
          born xR2 (+) born y in born x (+)born y by SURREALO:1,ORDINAL7:94;
          then reconsider xR1y=xR1*y,xR2y=xR2*y as Surreal by A15,A2;
          set BR = (born xR1 (+) born y) \/ (born xR2 (+) born y);
          A65: BR in born x (+)born y by A64, ORDINAL3:12;
          A66: born xR1 (+) born y c= BR & born xR2 (+) born y c= BR
          by XBOOLE_1:7;
          A67:xR1 == xR2 implies xR1y == xR2y by A66,A65,A15,A2;
          A68: yR in L_y \/ R_y & yL in L_y \/ R_y by A62,A63,XBOOLE_0:def 3;
          then  A69: born xR1 (+) born yL in born xR1 (+)born y &
          born xR1 (+) born yR in born xR1 (+)born y &
          born xR2 (+) born yL in born xR2 (+)born y &
          born xR2 (+) born yR in born xR2 (+)born y
          by SURREALO:1,ORDINAL7:94;
          born xR1 (+) born yL in born x (+)born y &
          born xR1 (+) born yR in born x (+)born y &
          born xR2 (+) born yR in born x (+)born y &
          born xR2 (+) born yL in born x (+)born y
          by A69,A64,ORDINAL1:10;
          then reconsider xR1yR=xR1*yR,xR2yR=xR2*yR,xR2yL=xR2*yL,xR1yL=xR1*yL
          as Surreal by A15,A2;
          set BY=(born x (+) born yL)\/(born x (+) born yR);
          A70: born x (+) born yL in born x (+)born y &
          born x (+) born yR in born x (+)born y
          by A68,SURREALO:1,ORDINAL7:94;
          then reconsider xyL=x*yL,xyR=x*yR as Surreal by A15,A2;
          A71: born xR1 (+) born yR c= BR & born xR2 (+) born yR c= BR &
          born xR2 (+) born yL c= BR &
          born xR1 (+) born yL c= BR by A69, A66,ORDINAL1:def 2;
          A72:xR1 == xR2 implies xR1yR == xR2yR by A71,A65, A15,A2;
          L_y << R_y by SURREAL0:45;
          then A73: yL < yR by A62,A63;
          x <=x & y <= y;
          then A74: L_x << {x} << R_x & x in {x} &
          L_y << {y} << R_y & y in {y} by SURREAL0:43,TARSKI:def 1;
          then A75: x<xR1 & x<xR2 & yL < y by A62,A63;
          A76: BY in born x (+)born y by A70,ORDINAL3:12;
          A77: born x (+) born yL c= BY & born x (+) born yR c= BY
          by XBOOLE_1:7;
          set BB = BR \/BY;
          A78: BB in born x (+)born y by A65,A76,ORDINAL3:12;
          A79:BR c= BB & BY c= BB by XBOOLE_1:7;
          then
          A80:born xR1 (+) born yL c= BB& born xR1 (+) born yR c= BB &
          born x (+) born yL c= BB & born x (+) born yR c= BB
          by A71,A77,XBOOLE_1:1;
          A81: born xR1 (+) born y c= BB & born xR2 (+) born y c= BB
          by A66,A79,XBOOLE_1:1;
          A82: born xR2 (+) born yL c= BB &
          born xR2 (+) born yR  c= BB by A71,A79,XBOOLE_1:1;
          per cases;
          suppose A83: xR1 < xR2;
            yL < y & xR1 < xR2 by A83,A74,A63;
            then xR2yL +xR1y < xR1yL+xR2y by A80,A81,A82,A78,A15,A2;
            then xR1y < xR1yL+xR2y - xR2yL by Th42;
            then xR1y < xR1yL+(xR2y - xR2yL) by Th37;
            then xR1y - xR1yL < xR2y - xR2yL by Th41;
            then (xR1y  - xR1yL)+xyL < xR2y - xR2yL +xyL by Th32;
            then (- xR1yL + xR1y)+xyL < - xR2yL+ (xR2y +xyL) by Th37;
            then A84: xR1y + xyL - xR1yL < xR2y + xyL - xR2yL by Th37;
            xyR + xR1yL < xyL + xR1yR by A73,A75,A80,A78,A15,A2;
            then xyR + xR1yL - xR1yR  < xyL by Th41;
            then xR1yL+ (xyR - xR1yR)  < xyL by Th37;
            then xyR +- xR1yR  < xyL -xR1yL by Th42;
            then xR1y + (xyR +- xR1yR)  < xR1y+(xyL +-xR1yL) by Th32;
            then (xR1y + xyR) +- xR1yR  < xR1y+(xyL +-xR1yL) by Th37;
            then xR1y + xyR + - xR1yR <= xR1y + xyL +- xR1yL by Th37;
            hence thesis by A62,A63,A84,SURREALO:4;
          end;
          suppose A85: xR2 < xR1;
            y < yR & xR2 < xR1 by A85,A74,A62;
            then xR1y + xR2yR < xR2y + xR1yR by A80,A81,A82,A78,A15,A2;
            then xR2yR +xR1y - xR1yR < xR2y by Th41;
            then xR2yR +(xR1y - xR1yR) < xR2y by Th37;
            then xR1y + - xR1yR  < xR2y -xR2yR by Th42;
            then xR1y + - xR1yR + xyR  < xR2y + -xR2yR + xyR by Th32;
            then A86: xR1y + xyR + - xR1yR < xR2y + -xR2yR + xyR by Th37;
            xyR + xR2yL < xyL +xR2yR by A73,A75,A78,A15,A2,A80,A82;
            then xyR < xR2yR + xyL - xR2yL by Th42;
            then xyR < xR2yR + (xyL - xR2yL) by Th37;
            then xyR -xR2yR < xyL - xR2yL by Th41;
            then xR2y +(-xR2yR + xyR) < xR2y +(xyL + - xR2yL) by Th32;
            then xR2y +-xR2yR + xyR < xR2y +(xyL + - xR2yL) by Th37;
            then xR2y + -xR2yR + xyR <= xR2y + xyL + - xR2yL by Th37;
            hence thesis by A62,A63,A86,SURREALO:4;
          end;
          suppose A87: xR1 == xR2;
            then - xR1yR <= - xR2yR by A72,Th10;
            then A88: (xR1y + xyR)+- xR1yR <= (xR1y + xyR)+- xR2yR by Th32;
            xyR + xR2yL < xyL +xR2yR by  A75,A73,A80,A82,A78,A15,A2;
            then xR2yL +  xyR - xR2yR < xyL by Th41;
            then xR2yL +  (xyR - xR2yR) < xyL by Th37;
            then xyR - xR2yR < xyL - xR2yL by Th42;
            then xR1y + (xyR - xR2yR) < xR2y+ (xyL - xR2yL) by A87,A67,Th44;
            then xR1y + (xyR - xR2yR) < (xR2y+ xyL) - xR2yL by
            Th37;
            then xR1y + xyR +- xR2yR < (xR2y+ xyL) +- xR2yL by Th37;
            hence l < r by A88,SURREALO:4,A62,A63;
          end;
        end;
        suppose  A89: l in comp(L_x,x,y,L_y) & r in comp(R_x,x,y,L_y);
          then consider xL,yL1 be Surreal such that
          A90:  l = (xL*y) +' (x*yL1) +' -' (xL*yL1) & xL in L_x & yL1 in L_y
          by Def14;
          consider xR,yL2 be Surreal such that
          A91:  r = (xR*y) +' (x*yL2) +' -' (xR*yL2) & xR in R_x & yL2 in L_y
          by Def14,A89;
          yL1 in L_y \/ R_y & yL2 in L_y \/ R_y by A90,A91,XBOOLE_0:def 3;
          then A92:born yL1 (+) born x in born x (+)born y &
          born yL2 (+) born x in born x (+)born y by SURREALO:1,ORDINAL7:94;
          then reconsider yL1x=yL1*x,yL2x=yL2*x as Surreal by A15,A2;
          set BL = (born yL1 (+) born x) \/ (born yL2 (+) born x);
          A93: BL in born x (+)born y by A92, ORDINAL3:12;
          A94: born yL1 (+) born x c= BL & born yL2 (+) born x c= BL by
          XBOOLE_1:7;
          A95:yL1 == yL2 implies yL1x == yL2x by A94,A93,A15,A2;
          A96: xR in L_x \/ R_x & xL in L_x \/ R_x by A90,A91,XBOOLE_0:def 3;
          A97: born yL1 (+) born xL in born yL1 (+)born x &
          born yL1 (+) born xR in born yL1 (+)born x &
          born yL2 (+) born xL in born yL2 (+)born x &
          born yL2 (+) born xR in born yL2 (+)born x
          by A96,SURREALO:1,ORDINAL7:94;
          then born yL1 (+) born xL in born x (+)born y &
          born yL1 (+) born xR in born x (+)born y &
          born yL2 (+) born xR in born x (+)born y &
          born yL2 (+) born xL in born x (+)born y
          by A92,ORDINAL1:10;
          then reconsider yL1xR=yL1*xR,yL2xR=yL2*xR,yL2xL=yL2*xL,yL1xL=yL1*xL
          as Surreal by A15,A2;
          set BY=(born y (+) born xL)\/(born y (+) born xR);
          A98: born y (+) born xL in born x (+)born y &
          born y (+) born xR in born x (+)born y
          by A96,SURREALO:1,ORDINAL7:94;
          then reconsider yxL=y*xL,yxR=y*xR as Surreal by A15,A2;
          A99: born yL1 (+) born xR c= BL & born yL2 (+) born xR c= BL &
          born yL2 (+) born xL c= BL &
          born yL1 (+) born xL c= BL by A97,A94,ORDINAL1:def 2;
          A100:yL1 == yL2 implies yL1xR == yL2xR by A99,A93,A15,A2;
          L_x << R_x by SURREAL0:45;
          then A101: xL < xR by A90,A91;
          x <=x & y <= y;
          then A102: L_x << {x} << R_x & x in {x} &
          L_y << {y} << R_y & y in {y} by SURREAL0:43,TARSKI:def 1;
          then A103: yL1<y & yL2<y & xL < x by A90,A91;
          A104: BY in born x (+)born y by A98,ORDINAL3:12;
          A105: born y (+) born xL c= BY & born y (+) born xR c= BY
          by XBOOLE_1:7;
          set BB = BL \/BY;
          A106: BB in born x (+)born y by A93,A104,ORDINAL3:12;
          A107: BL c= BB & BY c= BB by XBOOLE_1:7;
          then A108: born yL1 (+) born xL c= BB&
          born yL1 (+) born xR c= BB &
          born y (+) born xL c= BB & born y (+) born xR c= BB
          by A99,A105,XBOOLE_1:1;
          A109: born yL1 (+) born x c= BB & born yL2 (+) born x c= BB
          by A94,A107,XBOOLE_1:1;
          A110: born xL (+) born yL2 c= BB & born xR (+) born yL2 c= BB
          by A99,A107,XBOOLE_1:1;
          yL1xR+yxL < yL1xL+yxR by A106,A15,A2,A108,A101,A103;
          then yxL < yL1xL+yxR - yL1xR by Th42;
          then yxL < yL1xL+(yxR - yL1xR) by Th37;
          then A111: yxL - yL1xL < yxR - yL1xR by Th41;
          A112: xL*yL2 = yL2xL & x*yL2 = yL2x by A2,A106,A15,A109,A110;
          A113: xL*yL1 = yL1xL & x*yL1 = yL1x by A2,A106,A15,A108,A109;
          A114: xR*yL2 = yL2xR & xR*yL1 = yL1xR by A2,A106,A15,A108,A110;
          A115: l = (yxL) + (yL1x) + - (yL1xL) by A90,A2,A106,A15,A108,A113
          .= yL1x + yxL +- (yL1xL);
          A116: r = yxR + yL2x + - yL2xR by A91,A2,A106,A15,A108,A112,A114
          .= yL2x + yxR + - yL2xR;
          per cases;
          suppose A117: yL1 < yL2;
            yL2xL + yL1x < yL1xL + yL2x
            by A117,A103,A108,A109,A110,A106,A15,A2;
            then  yL1x < yL1xL + yL2x - yL2xL by Th42;
            then  yL1x < yL1xL +(yL2x  - yL2xL) by Th37;
            then  yL1x - yL1xL <  yL2x - yL2xL by Th41;
            then  A118: yL1x +- yL1xL + yxL <= yL2x - yL2xL + yxL by Th32;
            A119: yL2x - yL2xL + yxL =yxL - yL2xL + yL2x by Th37;
            yL2xR + yxL < yL2xL + yxR by A101,A103,A106,A15,A2,A108,A110;
            then yxL < yL2xL + yxR - yL2xR by Th42;
            then yxL < yL2xL + (yxR +- yL2xR) by Th37;
            then yxL - yL2xL < yxR +- yL2xR by Th41;
            then yxL - yL2xL + yL2x < yxR +- yL2xR +yL2x by Th32;
            then yL1x - yL1xL + yxL < yxR +- yL2xR +yL2x
            by SURREALO:4,A118,A119;
            then yL1x +(- yL1xL + yxL) < yxR - yL2xR +yL2x by Th37;
            then yL1x +(yxL - yL1xL) < yxR +(yL2x- yL2xR) by Th37;
            then yL1x +yxL - yL1xL < yxR +(yL2x- yL2xR) by Th37;
            hence l < r by A115,A116,Th37;
          end;
          suppose A120: yL2 < yL1;
            A121: born x (+) born yL1 c= BB & born xR (+) born yL1 c= BB
            by A99,XBOOLE_1:1,A94,A107;
            x < xR & yL2 < yL1 by A120,A102,A91;
            then yL1x + yL2xR < yL2x + yL1xR by A109,A110,A121,A106,A15,A2;
            then yL1x  < yL2x + yL1xR -yL2xR by Th42;
            then yL1x  < yL1xR+ (yL2x  -yL2xR) by Th37;
            then yL1x - yL1xR  <  yL2x  -yL2xR by Th41;
            then yL1x - yL1xR  +yxR <  yL2x  -yL2xR +yxR by Th32;
            then yL1x - yL1xR  +yxR <  yL2x  +yxR -yL2xR by Th37;
            then A122:yL1x + (yxR+- yL1xR) <  yL2x  +yxR+ -yL2xR by Th37;
            yL1xR + yxL < yL1xL + yxR by A101,A103,A106,A15,A2,A108;
            then yxL < yL1xL + yxR -yL1xR by Th42;
            then yxL < yL1xL + (yxR -yL1xR) by Th37;
            then yxL - yL1xL < yxR -yL1xR by Th41;
            then  yL1x +(yxL +- yL1xL) <= yL1x +(yxR -yL1xR) by Th32;
            then yL1x +(yxL +- yL1xL) < yL2x  +yxR+ -yL2xR
            by A122,SURREALO:4;
            hence l < r by A115,A116,Th37;
          end;
          suppose A123: yL1 == yL2;
            then - yL1xR <= - yL2xR by Th10,A100;
            then A124: (yL2x + yxR)+- yL1xR <= (yL2x + yxR)+- yL2xR by Th32;
            yL1x + (yxL +- yL1xL) < yL2x+(yxR +- yL1xR) by A123,A95,A111,Th44;
            then yL1x + (yxL +- yL1xL) < (yL2x +yxR) +- yL1xR by Th37;
            then yL1x + (yxL +- yL1xL) < (yL2x + yxR)+- yL2xR
            by A124,SURREALO:4;
            hence l < r by Th37,A115,A116;
          end;
        end;
        suppose A125: l in comp(R_x,x,y,R_y) & r in comp(L_x,x,y,R_y);
          then consider xR,yR1 be Surreal such that
          A126:  l = (xR*y) +' (x*yR1) +' -' (xR*yR1) & xR in R_x & yR1 in R_y
          by Def14;
          consider xL,yR2 be Surreal such that
          A127:  r = (xL*y) +' (x*yR2) +' -' (xL*yR2) & xL in L_x & yR2 in R_y
          by Def14,A125;
          yR1 in L_y \/ R_y & yR2 in L_y \/ R_y by A126,A127,XBOOLE_0:def 3;
          then A128:born yR1 (+) born x in born y (+)born x &
          born yR2 (+) born x in born y (+)born x by SURREALO:1,ORDINAL7:94;
          then reconsider yR1x=yR1*x,yR2x=yR2*x as Surreal by A15,A2;
          set BR = (born yR1 (+) born x) \/ (born yR2 (+) born x);
          A129: BR in born x (+)born y by A128, ORDINAL3:12;
          A130: born yR1 (+) born x c= BR & born yR2 (+) born x c= BR
          by XBOOLE_1:7;
          A131:yR1 == yR2 implies yR1x == yR2x by A130,A129,A15,A2;
          A132: xR in L_x \/ R_x & xL in L_x \/ R_x
          by A126,A127,XBOOLE_0:def 3;
          A133:  born yR1 (+) born xL in born yR1 (+)born x &
          born yR1 (+) born xR in born yR1 (+)born x &
          born yR2 (+) born xL in born yR2 (+)born x &
          born yR2 (+) born xR in born yR2 (+)born x
          by A132,SURREALO:1,ORDINAL7:94;
          born yR1 (+) born xL in born x (+)born y &
          born yR1 (+) born xR in born x (+)born y &
          born yR2 (+) born xR in born x (+)born y &
          born yR2 (+) born xL in born x (+)born y
          by A128,ORDINAL1:10,A133;
          then reconsider yR1xR=yR1*xR,yR2xR=yR2*xR,yR2xL=yR2*xL,
          yR1xL=yR1*xL as Surreal by A15,A2;
          set BY=(born y (+) born xL)\/(born y (+) born xR);
          A134: born y (+) born xL in born x (+)born y &
          born y (+) born xR in born x (+)born y
          by A132,SURREALO:1,ORDINAL7:94;
          then reconsider yxL=y*xL,yxR=y*xR as Surreal by A15,A2;
          A135: born yR1 (+) born xR c= BR & born yR2 (+) born xR c= BR &
          born yR2 (+) born xL c= BR &
          born yR1 (+) born xL c= BR by A133,A130,ORDINAL1:def 2;
          A136:yR1 == yR2 implies yR1xR == yR2xR by A135,A129,A15,A2;
          L_x << R_x by SURREAL0:45;
          then A137: xL < xR by A126,A127;
          x <=x & y <= y;
          then A138: L_x << {x} << R_x & x in {x} &
          L_y << {y} << R_y & y in {y} by SURREAL0:43,TARSKI:def 1;
          then
          A139: y<yR1 & y<yR2 & xL < x by A126,A127;
          A140: BY in born x (+)born y by A134,ORDINAL3:12;
          A141: born y (+) born xL c= BY & born y (+) born xR c= BY
          by XBOOLE_1:7;
          set BB = BR \/BY;
          A142:BB in born x (+)born y by A129,A140,ORDINAL3:12;
          A143:BR c= BB & BY c= BB by XBOOLE_1:7;
          then
          A144: born yR1 (+) born xL c= BB&
          born yR1 (+) born xR c= BB &
          born y (+) born xL c= BB &
          born y (+) born xR c= BB by A135,A141,XBOOLE_1:1;
          A145: born yR1 (+) born x c= BB & born yR2 (+) born x c= BB
          by A130,A143,XBOOLE_1:1;
          A146: born yR2 (+) born xL c= BB &
          born yR2 (+) born xR  c= BB by A135,A143,XBOOLE_1:1;
          A147: xR*y = yxR & xL*y = yxL by A2,A142,A15,A144;
          A148: xL*yR2 = yR2xL  & x*yR2 = yR2x by A2,A142,A15,A145,A146;
          A149: xR*yR2 = yR2xR & xR*yR1 = yR1xR by A2,A142,A15,A144,A146;
          A150: l = yxR + yR1x + - yR1xR by A126,A147,A2,A142,A15,A145,A149
          .= yR1x +yxR +-yR1xR;
          A151: r =yxL + yR2x + - yR2xL by A127,A2,A142,A15,A144,A148
          .= yR2x + yxL + - yR2xL;
          per cases;
          suppose A152: yR1 < yR2;
            yR2xL +yR1x < yR1xL+yR2x by A152,A139,A144,A145,A146,A142,A15,A2;
            then yR1x < yR1xL+yR2x - yR2xL by Th42;
            then yR1x < yR1xL+(yR2x - yR2xL) by Th37;
            then yR1x - yR1xL < yR2x - yR2xL by Th41;
            then (yR1x  - yR1xL)+yxL < yR2x - yR2xL +yxL by Th32;
            then (- yR1xL + yR1x)+yxL < - yR2xL+ (yR2x +yxL) by Th37;
            then A153: yR1x + yxL +- yR1xL < yR2x + yxL - yR2xL by Th37;
            yxR + yR1xL < yxL + yR1xR by A137,A139,A144,A142,A15,A2;
            then yxR + yR1xL - yR1xR  < yxL by Th41;
            then yR1xL+ (yxR - yR1xR) < yxL by Th37;
            then yxR +- yR1xR  < yxL -yR1xL by Th42;
            then yR1x + (yxR - yR1xR) < yR1x+(yxL -yR1xL) by Th32;
            then (yR1x + yxR) - yR1xR < yR1x+(yxL -yR1xL) by Th37;
            then yR1x + yxR  - yR1xR <= yR1x + yxL - yR1xL by Th37;
            hence thesis by A151,A150,A153,SURREALO:4;
          end;
          suppose A154: yR2 < yR1;
            x < xR & yR2 < yR1 by A154,A138,A126;
            then yR1x + yR2xR < yR2x + yR1xR by
            A144,A145,A146,A142,A15,A2;
            then yR1x + yR2xR - yR1xR < yR2x by Th41;
            then yR2xR +(yR1x - yR1xR) < yR2x by Th37;
            then yR1x - yR1xR < yR2x -yR2xR by Th42;
            then yR1x - yR1xR + yxR < yR2x -yR2xR + yxR by Th32;
            then A155: yR1x + yxR + - yR1xR < yR2x -yR2xR + yxR by Th37;
            yxR + yR2xL < yxL +yR2xR by A137,A139,A142,A15,A2,A144,A146;
            then yxR < yR2xR + yxL - yR2xL by Th42;
            then yxR < yR2xR + (yxL - yR2xL) by Th37;
            then yxR-yR2xR  < yxL  - yR2xL by Th41;
            then yR2x +(-yR2xR + yxR) < yR2x +(yxL - yR2xL) by Th32;
            then yR2x +-yR2xR + yxR < yR2x +(yxL - yR2xL) by Th37;
            then yR2x -yR2xR + yxR <= yR2x + yxL - yR2xL by Th37;
            hence thesis by A151,A150,A155,SURREALO:4;
          end;
          suppose A156: yR1 == yR2;
            then - yR1xR <= - yR2xR by A136,Th10;
            then A157: (yR1x + yxR)- yR1xR <= (yR1x + yxR)- yR2xR by Th32;
            yxR + yR2xL < yxL +yR2xR by A139,A137,A144,A146,A142,A15,A2;
            then yR2xL +  yxR - yR2xR < yxL by Th41;
            then yR2xL +  (yxR - yR2xR) < yxL by Th37;
            then yxR - yR2xR < yxL - yR2xL by Th42;
            then yR1x + (yxR - yR2xR) < yR2x+ (yxL - yR2xL)
            by A156,A131,Th44;
            then yR1x + (yxR - yR2xR) < (yR2x+ yxL) - yR2xL by Th37;
            then yR1x + yxR - yR2xR < (yR2x+ yxL) - yR2xL by Th37;
            hence l < r by A151,A150,A157,SURREALO:4;
          end;
        end;
      end;
      then [(comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y)),
      (comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y))] in Day succ U
      by A25,SURREAL0:46;
      hence thesis by Th50;
    end;
    defpred Q24[Ordinal] means
    for x1,x2,y be Surreal st born x1 (+) born x2 (+) born y c= $1
      holds P2[D,x1,x2,y] & P4[D,x1,x2,y];
    A158: for E be Ordinal st for C be Ordinal st C in E holds Q24[C]
      holds Q24[E]
    proof
      let E be Ordinal such that A159: for C be Ordinal st C in E holds Q24[C];
      A160: for x1,x2,y,x1y,x2y be Surreal st
         born x1 (+) born x2 (+) born y c= E &
         born x1 (+) born y c= D & born x2 (+) born y c= D &
         x1 == x2 & x1y = x1*y & x2y = x2*y
      holds L_x1y << {x2y} << R_x1y
      proof
        let x1,x2,y,x1y,x2y be Surreal such that
        A161: born x1 (+) born x2 (+) born y c= E
        and
        A162: born x1 (+) born y c= D & born x2 (+) born y c= D and
        A163: x1 == x2 & x1y = x1*y & x2y = x2*y;
        A164:x1y =[(comp(L_x1,x1,y,L_y) \/ comp(R_x1,x1,y,R_y)),
        (comp(L_x1,x1,y,R_y) \/ comp(R_x1,x1,y,L_y))] by A163,Th50;
        A165:  L_y << {y} << R_y & y in {y} by SURREALO:11,TARSKI:def 1;
        thus L_x1y << {x2y}
        proof
          let l,r such that A166: l in L_x1y & r in {x2y};
          per cases by A164,A166,XBOOLE_0:def 3;
          suppose
            l in comp(R_x1,x1,y,R_y);
            then consider x3,y3 be Surreal such that
            A167: l = (x3*y) +' (x1*y3) +' -' (x3*y3) & x3 in R_x1 & y3 in R_y
            by Def14;
            A168: {x1} << R_x1 & x1 in {x1} by SURREALO:11,TARSKI:def 1;
            A169: y3 in L_y \/ R_y by A167,XBOOLE_0:def 3;
            then A170:born x1 (+) born y3 in born x1 (+)born y &
            born x2 (+) born y3 in born x2 (+)born y
            by SURREALO:1,ORDINAL7:94;
            then reconsider x1y3=x1*y3,x2y3=x2*y3 as Surreal by A2,A162;
            set BL = (born x1 (+) born y3) \/ (born x2 (+) born y3);
            x3 in L_x1 \/R_x1 by A167,XBOOLE_0:def 3;
            then A171: born x3 (+) born y3 in born x1 (+) born y3 &
            born x3 (+) born y in born x1 (+) born y
            by SURREALO:1,ORDINAL7:94;
            then A172: born x3 (+) born y3 in D &
            born x3 (+) born y in D by A170,A162,ORDINAL1:10;
            then reconsider x3y3=x3*y3,x3y=x3*y as Surreal by A2;
            A173: (born x1 (+) born x2) (+) born y3 in
            (born x1 (+) born x2) (+) born y
            by A169,SURREALO:1,ORDINAL7:94;
            born x1 (+) born y3 c= D & born x2 (+) born y3 c= D
            by A170,A162,ORDINAL1:def 2;
            then A174: x1y3 == x2y3 by A173,A163,A161,A159;
            born y (+) born x3 (+) born x2 in born y (+) born x1 (+) born x2
            by A171,ORDINAL7:94;
            then A175: born y (+) born x3 (+) born x2 in
            born y (+) (born x1 (+) born x2) by ORDINAL7:68;
            A176: born y (+) born x3 (+) born x2 =
            born y (+) born x2 (+) born x3 by ORDINAL7:68;
            A177:born x2 (+) born y3 c= D by A170,A162,ORDINAL1:def 2;
            A178: y3*x2 = x2y3 by A3,A170,A162,ORDINAL1:def 2;
            A179:born x3 (+) born y c= D & born x2 (+) born y c= D
            by A171,A162,ORDINAL1:def 2;
            A180: y*x3 = x3y & y*x2 = x2y by A162,A163,A3,A171,A2;
            A181:born x3 (+) born y3 c= D
            by A171,A170,A162,ORDINAL1:10,ORDINAL1:def 2;
            A182: y3*x3 = x3y3 by A172,A2;
            y < y3 & x2 < x3 by A168,A167,A163,SURREALO:4,A165;
            then x2y3 + x3y < x3y3 + x2y
            by A167,A176,A175,A161,A159,A177,A178,A179,A180,A181,A182;
            then A183:  x2y3 + x3y -x3y3 < x2y by Th41;
            x1y3 +(x3y +-x3y3) <= x2y3 +(x3y -x3y3) by A174,Th32;
            then x1y3 +x3y -x3y3 <= x2y3 +(x3y -x3y3) by Th37;
            then x1y3 +x3y -x3y3 <= x2y3 +x3y -x3y3 by Th37;
            then A184: x1y3 +x3y -x3y3 < x2y by A183,SURREALO:4;
            l = x3y+x1y3-x3y3 by A167;
            hence thesis by A166,A184,TARSKI:def 1;
          end;
          suppose l in comp(L_x1,x1,y,L_y);
            then consider x3,y3 be Surreal such that
            A185: l = (x3*y) +' (x1*y3) +' -' (x3*y3) & x3 in L_x1 & y3 in L_y
            by Def14;
            A186: L_x1 << {x1} & x1 in {x1} by SURREALO:11,TARSKI:def 1;
            A187: y3 in L_y \/ R_y by A185,XBOOLE_0:def 3;
            then A188:born x1 (+) born y3 in born x1 (+)born y &
            born x2 (+) born y3 in born x2 (+)born y
            by SURREALO:1,ORDINAL7:94;
            then reconsider x1y3=x1*y3,x2y3=x2*y3 as Surreal by A2,A162;
            set BL = (born x1 (+) born y3) \/ (born x2 (+) born y3);
            x3 in L_x1 \/R_x1 by A185,XBOOLE_0:def 3;
            then A189: born x3 (+) born y3 in born x1 (+) born y3 &
            born x3 (+) born y in born x1 (+) born y
            by SURREALO:1,ORDINAL7:94;
            then A190: born x3 (+) born y3 in D & born x3 (+) born y in D
            by A188,A162,ORDINAL1:10;
            then reconsider x3y3=x3*y3,x3y=x3*y as Surreal by A2;
            A191: born x1 (+) born x2 (+) born y3 in
            born x1 (+) born x2 (+) born y
            by A187,SURREALO:1,ORDINAL7:94;
            born x1 (+) born y3 c= D & born x2 (+) born y3 c= D
            by A188,A162,ORDINAL1:def 2;
            then A192: x1y3 == x2y3 by A191,A161,A159,A163;
            born y (+) born x3 (+) born x2 in born y (+) born x1 (+) born x2
            by A189,ORDINAL7:94;
            then A193: born y (+) born x3 (+) born x2 in
            born y (+) (born x1 (+) born x2) by ORDINAL7:68;
            A194:born x2 (+) born y3 c= D by A188,A162,ORDINAL1:def 2;
            A195: y3*x2 = x2y3 by A3,A188,A162,ORDINAL1:def 2;
            A196:born x3 (+) born y c= D & born x2 (+) born y c= D
            by A189,A162,ORDINAL1:def 2;
            A197: y*x3 = x3y & y*x2 = x2y
            by A163,A3,A162,A189,ORDINAL1:def 2;
            A198:born x3 (+) born y3 c= D
            by A188,A162,ORDINAL1:10,A189,ORDINAL1:def 2;
            A199: y3*x3 = x3y3 by A190,A2;
            A200: y3<y & x3 < x2 by A186,A185,A163,SURREALO:4,A165;
            x2y3 + x3y < x3y3 + x2y
            by A185,A200,A194,A195,A196,A197,A198,A199,A193,A161,A159;
            then A201:  x2y3 + x3y -x3y3 < x2y by Th41;
            x1y3 +(x3y +-x3y3) <= x2y3 +(x3y +-x3y3) by A192,Th32;
            then x1y3 +x3y +-x3y3 <= x2y3 +(x3y +-x3y3) by Th37;
            then x1y3 +x3y +-x3y3 <= x2y3 +x3y +-x3y3 by Th37;
            then A202: x1y3 +x3y +-x3y3 < x2y by A201,SURREALO:4;
            l = x3y+x1y3+-x3y3 by A185;
            hence thesis by A166,A202,TARSKI:def 1;
          end;
        end;
        thus {x2y} << R_x1y
        proof
          let l,r such that A203: l in {x2y} & r in R_x1y;
          per cases by XBOOLE_0:def 3,A164,A203;
          suppose
            r in comp(L_x1,x1,y,R_y);
            then consider x3,y3 be Surreal such that
            A204: r = (x3*y) +' (x1*y3) +' -' (x3*y3) &
            x3 in L_x1 & y3 in R_y by Def14;
            A205: L_x1 << {x1} & x1 in {x1} by SURREALO:11,TARSKI:def 1;
            A206:y3 in L_y \/ R_y by A204,XBOOLE_0:def 3;
            then A207:born x1 (+) born y3 in born x1 (+)born y &
            born x2 (+) born y3 in born x2 (+)born y
            by SURREALO:1,ORDINAL7:94;
            then reconsider x1y3=x1*y3,x2y3=x2*y3 as Surreal by A162,A2;
            set BL = (born x1 (+) born y3) \/ (born x2 (+) born y3);
            x3 in L_x1 \/R_x1 by A204,XBOOLE_0:def 3;
            then A208: born x3 (+) born y3 in born x1 (+) born y3 &
            born x3 (+) born y in born x1 (+) born y
            by SURREALO:1,ORDINAL7:94;
            then A209:born x3 (+) born y3 in D &
            born x3 (+) born y in D
            by A207,A162,ORDINAL1:10;
            then reconsider x3y3=x3*y3,x3y=x3*y as Surreal by A2;
            A210:  born x1 (+) born x2 (+) born y3 in
            born x1 (+) born x2 (+) born y
            by A206,SURREALO:1,ORDINAL7:94;
            born x1 (+) born y3 c= D & born x2 (+) born y3 c= D
            by A207,A162,ORDINAL1:def 2;
            then A211: x1y3 == x2y3 by A210,A161,A159,A163;
            born y (+) born x3 (+) born x2 in born y (+) born x1 (+) born x2
            by A208,ORDINAL7:94;
            then A212: born y (+) born x3 (+) born x2 in
            born y (+) (born x1 (+) born x2) by ORDINAL7:68;
            A213:born x2 (+) born y3 c= D by A207,A162,ORDINAL1:def 2;
            A214: y3*x2 = x2y3 by A3,A207,A162,ORDINAL1:def 2;
            A215:born x3 (+) born y c= D & born x2 (+) born y c= D
            by A208,A162,ORDINAL1:def 2;
            A216: y*x3 = x3y & y*x2 = x2y by A208,A162,ORDINAL1:def 2,A163,A3;
            A217:born x3 (+) born y3 c= D
            by A207,A162,ORDINAL1:10, A208,ORDINAL1:def 2;
            A218: y3*x3 = x3y3 by A209,A2;
            A219: y < y3 & x3 < x2 by A205,A204,A163,SURREALO:4,A165;
            x3y3 + x2y < x2y3 + x3y
            by A204,A212,A161,A159,A219,A213,A214,A215,A216,A217,A218;
            then A220:  x2y < x2y3 + x3y -x3y3 by Th42;
            x2y3 +(x3y +-x3y3) <= x1y3 +(x3y +-x3y3) by A211,Th32;
            then x2y3 +x3y +-x3y3 <= x1y3 +(x3y +-x3y3) by Th37;
            then x2y3 +x3y +-x3y3 <= x1y3 +x3y +-x3y3 by Th37;
            then A221: x2y < x1y3 +x3y +-x3y3 by A220,SURREALO:4;
            r = x3y+x1y3+-x3y3 by A204;
            hence thesis by A203,A221,TARSKI:def 1;
          end;
          suppose r in comp(R_x1,x1,y,L_y);
            then consider x3,y3 be Surreal such that
            A222: r = (x3*y) +' (x1*y3) +' -' (x3*y3) & x3 in R_x1 & y3 in L_y
            by Def14;
            A223: {x1} << R_x1 &
            x1 in {x1} by SURREALO:11,TARSKI:def 1;
            A224: y3 in L_y \/ R_y by A222,XBOOLE_0:def 3;
            then A225:born x1 (+) born y3 in born x1 (+)born y &
            born x2 (+) born y3 in born x2 (+)born y
            by SURREALO:1,ORDINAL7:94;
            then reconsider x1y3=x1*y3,x2y3=x2*y3 as Surreal by A2,A162;
            set BL = (born x1 (+) born y3) \/ (born x2 (+) born y3);
            x3 in L_x1 \/R_x1 by A222,XBOOLE_0:def 3;
            then A226: born x3 (+) born y3 in born x1 (+) born y3 &
            born x3 (+) born y in born x1 (+) born y
            by SURREALO:1,ORDINAL7:94;
            then A227:born x3 (+) born y3 in D & born x3 (+) born y in D
            by A225,A162,ORDINAL1:10;
            then reconsider x3y3=x3*y3,x3y=x3*y as Surreal by A2;
            A228:(born x1 (+) born x2) (+) born y3 in
            (born x1 (+) born x2) (+) born y
            by A224,SURREALO:1,ORDINAL7:94;
            born x1 (+) born y3 c= D & born x2 (+) born y3 c= D
            by A225,A162,ORDINAL1:def 2;
            then A229: x1y3 == x2y3 by A228,A159,A161,A163;
            born y (+) born x3 (+) born x2 in born y (+) born x1 (+) born x2
            by A226,ORDINAL7:94;
            then A230: born y (+) born x3 (+) born x2 in
            born y (+) (born x1 (+) born x2) by ORDINAL7:68;
            A231: born y (+) born x3 (+) born x2
            = born y (+) born x2 (+) born x3 by ORDINAL7:68;
            A232:born x2 (+) born y3 c= D by A225,A162,ORDINAL1:def 2;
            A233: y3*x2 = x2y3 by A3,A225,A162,ORDINAL1:def 2;
            A234:born x3 (+) born y c= D & born x2 (+) born y c= D
            by A226,A162,ORDINAL1:def 2;
            A235: y*x3 = x3y & y*x2 = x2y by A163,A3,A226,A162,ORDINAL1:def 2;
            A236:born x3 (+) born y3 c= D by A226,A225,A162,ORDINAL1:10,
            ORDINAL1:def 2;
            A237: y3*x3 = x3y3 by A3,A227,ORDINAL1:def 2;
            A238: y3<y & x2 < x3 by A223,A222,A163,SURREALO:4,A165;
            x3y3 + x2y < x2y3 + x3y
            by A231,A222,A230,A159,A161,A238,A232,A233,A234,A235,A236,A237;
            then A239:  x2y < x2y3 + x3y -x3y3 by Th42;
            x2y3 +(x3y +-x3y3) <= x1y3 +(x3y -x3y3) by A229,Th32;
            then x2y3 +x3y -x3y3 <= x1y3 +(x3y -x3y3) by Th37;
            then x2y3 +x3y -x3y3 <= x1y3 +x3y -x3y3 by Th37;
            then A240: x2y < x1y3 +x3y -x3y3 by A239,SURREALO:4;
            r = x3y+x1y3-x3y3 by A222;
            hence thesis by A203,A240,TARSKI:def 1;
          end;
        end;
      end;
      let x,y1,y2 be Surreal such that
      A241: born x (+) born y1 (+) born y2 c= E;
      thus P2[D,x,y1,y2]
      proof
        let x1y,x2y be Surreal such that
        A242:   born x (+) born y2 c= D & born y1 (+) born y2 c= D &
        x == y1 & x1y = x*y2 & x2y = y1*y2;
        A243: L_x1y << {x2y} << R_x1y & L_x2y << {x1y} << R_x2y
        by A241,A242,A160;
        then reconsider z= [L_x1y \/ L_x2y, R_x1y \/ R_x2y] as Surreal
        by SURREALO:14;
        x1y == z == x2y by A243,SURREALO:15;
        hence x1y == x2y by SURREALO:4;
      end;
      assume y1 < y2;
      then per cases by SURREALO:13;
      suppose ex y2L be Surreal st y2L in L_y2 & y1 <= y2L < y2;
        then consider y2L be Surreal such that
        A244: y2L in L_y2 & y1 <= y2L < y2;
        A245:for xL be Surreal st xL in L_x holds P3[D,xL,x,y1,y2]
        proof
          let xL be Surreal such that A246: xL in L_x;
          let xLy2,xy1,xLy1,xy2 be Surreal such that
          A247:born xL (+) born y1 c= D & born x (+) born y1 c= D &
          born xL (+) born y2 c= D & born x (+) born y2 c= D and
          A248: xLy1 = xL*y1 & xLy2=xL*y2 & xy1=x*y1 & xy2=x*y2 and
          A249: xL < x & y1 < y2;
          A250: xy2 = [comp(L_x,x,y2,L_y2) \/ comp(R_x,x,y2,R_y2),
          comp(L_x,x,y2,R_y2) \/ comp(R_x,x,y2,L_y2)] by A248,Th50;
          A251: L_xy2  << {xy2} << R_xy2 & xy2 in {xy2}
          by SURREALO:11,TARSKI:def 1;
          A252: y2L in L_y2 \/ R_y2 by A244,XBOOLE_0:def 3;
          then A253: born x (+) born y2L in born x (+) born y2
          by SURREALO:1,ORDINAL7:94;
          then reconsider xy2L=x*y2L as Surreal by A2,A247;
          A254: xL in L_x \/ R_x by A246,XBOOLE_0:def 3;
          then A255:born xL (+) born y2L in born x (+) born y2L
          by SURREALO:1,ORDINAL7:94;
          then A256: born xL (+) born y2L in D by A247,A253,ORDINAL1:10;
          then reconsider xLy2L=xL*y2L as Surreal by A2;
          (xLy2) +' xy2L +' -' xLy2L in comp(L_x,x,y2,L_y2)
          by A246,A244,A248,Def14;
          then (xLy2) +' xy2L +' -' xLy2L in L_xy2 by A250,XBOOLE_0:def 3;
          then xLy2 + xy2L - xLy2L < xy2 by A251;
          then A257: xLy2 + xy2L < xy2 + xLy2L by Th41;
          A258: born y1 (+) born y2 (+) born x = born x (+) born y1 (+) born y2
          by ORDINAL7:68;
          A259: born y2L (+) born x c= D by A253,A247,ORDINAL1:def 2;
          A260: y2L*x = xy2L & y1*x = xy1
          by A253,A247,A248,A3,ORDINAL1:def 2;
          A261:  born y2L (+) born xL c= D
          by A255,A247,A253,ORDINAL1:10,ORDINAL1:def 2;
          A262: y2L*xL  = xLy2L by A3,A256,ORDINAL1:def 2;
          A263: y1*xL =xLy1 by A247,A248,A3;
          A264: born y2L (+) born y1 in born y1 (+) born y2
          by A252,SURREALO:1,ORDINAL7:94;
          then A265: born y2L (+) born y1 (+) born x in
          born y2 (+) born y1 (+) born x by ORDINAL7:94;
          then A266: born y2L (+) born y1 (+) born x in E by A241,A258;
          born y2L (+) born y1 (+) born xL in born y2 (+) born y1 (+) born xL &
          born y2 (+) born y1 (+) born xL in
          born y2 (+) born y1 (+) born x by A254,SURREALO:1,A264,ORDINAL7:94;
          then
          A267: born y2L (+) born y1 (+) born xL in
          born y1 (+) born y2 (+) born x by ORDINAL1:10;
          per cases by A244;
          suppose A268:y1 < y2L;
            born x (+) born y1 (+) born y2L in E by A266,ORDINAL7:68;
            then A269: xLy2L+xy1 <= xLy1+xy2L
            by A246,A159,A249,A268,A261,A247,A248,A259;
            A270: xLy2L+xy1 +(xLy2 + xy2L) < xLy1+xy2L +(xy2 + xLy2L)
            by A257,A269,Th44;
            A271: xLy2L+xy1 +(xLy2 + xy2L) = xLy2L+(xy1 +(xLy2 + xy2L))
            by Th37
            .= xLy2L+((xy1 +xLy2) + xy2L) by Th37
            .= (xLy2L + xy2L) + (xLy2+ xy1) by Th37;
            xLy1+xy2L +(xy2 + xLy2L) = xy2L + (xLy1 +(xy2 + xLy2L)) by Th37
            .= xy2L + ((xLy1 +xy2) + xLy2L) by Th37
            .= (xLy2L + xy2L) + (xLy1+xy2) by Th37;
            hence xLy2+ xy1 < xLy1+xy2 by A270,A271,Th43;
          end;
          suppose A272: y1 == y2L;
            A273: xy2L == xy1 by A241,A258,A265,A159,A247,A259,A272,A260;
            A274: xLy2L == xLy1
            by A261,A263,A262,A247,A272,A267,A159,A241,A258;
            A275: xy2 + xLy2L <= xLy1+xy2 by A274,Th32;
            A276:xLy2 + xy1 <= xLy2 + xy2L by A273,Th32;
            xLy2 + xy1 < xy2 + xLy2L by A257,A276,SURREALO:4;
            hence xLy2+ xy1 < xLy1+xy2 by A275,SURREALO:4;
          end;
        end;
        for xR be Surreal st xR in R_x holds P3[D,x,xR,y1,y2]
        proof
          let xR be Surreal such that A277: xR in R_x;
          let xy2,xRy1,xy1,xRy2 be Surreal such that
          A278:born x (+) born y1 c= D & born xR (+) born y1 c= D &
          born x (+) born y2 c= D & born xR (+) born y2 c= D and
          A279: xy1 = x*y1 & xy2=x*y2 & xRy1=xR*y1 & xRy2=xR*y2 and
          A280: x < xR & y1 < y2;
          A281: xy2 = [comp(L_x,x,y2,L_y2) \/ comp(R_x,x,y2,R_y2),
          comp(L_x,x,y2,R_y2) \/ comp(R_x,x,y2,L_y2)] by A279,Th50;
          A282: L_xy2  << {xy2} << R_xy2 & xy2 in {xy2}
          by SURREALO:11,TARSKI:def 1;
          A283: y2L in L_y2 \/ R_y2 by A244,XBOOLE_0:def 3;
          A284: born x (+) born y2L in born x (+) born y2
          by A283,SURREALO:1,ORDINAL7:94;
          reconsider xy2L=x*y2L as Surreal by A2,A284,A278;
          A285: xR in L_x \/ R_x by A277,XBOOLE_0:def 3;
          A286: born  xR (+) born y2L in born x (+) born y2L
          by A285,SURREALO:1,ORDINAL7:94;
          then A287: born xR (+) born y2L in D by A278,A284,ORDINAL1:10;
          then reconsider xRy2L=xR*y2L as Surreal by A2;
          (xRy2) +' xy2L +' -' xRy2L in comp(R_x,x,y2,L_y2)
          by A277,A244,A279,Def14;
          then (xRy2) +' xy2L +' -' xRy2L in R_xy2 by A281,XBOOLE_0:def 3;
          then A288: xy2 < (xRy2) + xy2L - xRy2L by A282;
          then A289: xy2 + xRy2L < (xRy2) + xy2L by Th42;
          A290: born y1 (+) born y2 (+) born x = born x (+) born y1 (+) born y2
          by ORDINAL7:68;
          A291: born y2L (+) born x c= D by A284,A278,ORDINAL1:def 2;
          A292: y2L*x = xy2L & y1*x = xy1
          by  A284,A278,A279,A3,ORDINAL1:def 2;
          A293:  born y2L (+) born xR c= D by A286,A278,A284,ORDINAL1:10,
          ORDINAL1:def 2;
          A294: y2L*xR  = xRy2L by A3,A287,ORDINAL1:def 2;
          A295: y1*xR =xRy1 by A278,A279,A3;
          A296: born y2L (+) born y1 in born y1 (+) born y2
          by A283,SURREALO:1,ORDINAL7:94;
          then A297: born y2L (+) born y1 (+) born x in
          born y1 (+) born y2 (+) born x by ORDINAL7:94;
          then A298: born y2L (+) born y1 (+) born x in E by A241,A290;
          born y2L (+) born y1 (+) born xR in
          born y2 (+) born y1 (+) born xR &
          born y2 (+) born y1 (+) born xR in
          born y2 (+) born y1 (+) born x by A285,SURREALO:1,A296,ORDINAL7:94;
          then A299:born y2L (+) born y1 (+) born xR in
          born y1 (+) born y2 (+) born x by ORDINAL1:10;
          per cases by A244;
          suppose A300:y1 < y2L;
            born x (+) born y1 (+) born y2L in E by A298,ORDINAL7:68;
            then A301: xy2L+xRy1 < xy1+xRy2L
            by A277,A159,A280,A300,A293,A278,A279,A291;
            xy2 + xRy2L <= (xRy2) + xy2L by Th42,A288;
            then A302: xy2 + xRy2L +(xy2L+xRy1) < (xRy2) + xy2L +(xy1+xRy2L)
            by A301,Th44;
            A303: xy2 + xRy2L +(xy2L+xRy1) = xy2 + (xRy2L +(xy2L+xRy1))
            by Th37
            .= xy2 + ((xRy2L +xy2L)+xRy1) by Th37
            .= xy2+ xRy1 + (xRy2L +xy2L) by Th37;
            xRy2 + xy2L +(xy1+xRy2L) = xy2L +(xRy2+(xy1+xRy2L)) by Th37
            .= xy2L +((xRy2+xy1)+xRy2L) by Th37
            .= xy1+xRy2 +(xRy2L +xy2L) by Th37;
            hence xy2+ xRy1 < xy1+xRy2 by A303,A302,Th43;
          end;
          suppose A304: y1 == y2L;
            A305: xy1 == xy2L by A159,A297,A241,A290,A278,A291,A304,A292;
            A306: xRy1 == xRy2L
            by A293,A295,A294,A278,A304,A299,A159,A241,A290;
            A307: xy2 + xRy1 <= xy2 + xRy2L by A306,Th32;
            A308: xRy2 + xy2L <= xy1+xRy2 by A305,Th32;
            xy2+ xRy1 < xRy2 + xy2L by A289,A307,SURREALO:4;
            hence xy2+ xRy1 < xy1+xRy2 by A308,SURREALO:4;
          end;
        end;
        hence thesis by A245;
      end;
      suppose ex y1R be Surreal st y1R in R_y1 & y1 < y1R <= y2;
        then consider y1R be Surreal such that
        A309: y1R in R_y1 & y1 < y1R <= y2;
        A310:  for xL be Surreal st xL in L_x holds P3[D,xL,x,y1,y2]
        proof
          let xL be Surreal such that A311: xL in L_x;
          let xLy2,xy1,xLy1,xy2 be Surreal such that
          A312:born xL (+) born y1 c= D & born x (+) born y1 c= D &
          born xL (+) born y2 c= D & born x (+) born y2 c= D and
          A313: xLy1 = xL*y1 & xLy2=xL*y2 & xy1=x*y1 & xy2=x*y2 and
          A314: xL < x & y1 < y2;
          A315: xy1 = [comp(L_x,x,y1,L_y1) \/ comp(R_x,x,y1,R_y1),
          comp(L_x,x,y1,R_y1) \/ comp(R_x,x,y1,L_y1)] by A313,Th50;
          A316: L_xy1  << {xy1} << R_xy1 & xy1 in {xy1}
          by SURREALO:11,TARSKI:def 1;
          A317: y1R in L_y1 \/ R_y1 by A309,XBOOLE_0:def 3;
          A318: born x (+) born y1R in born x (+) born y1
          by A317,SURREALO:1,ORDINAL7:94;
          reconsider xy1R=x*y1R as Surreal by A2,A312,A318;
          A319: xL in L_x \/ R_x by A311,XBOOLE_0:def 3;
          then A320: born xL (+) born y1R in born x (+) born y1R
          by SURREALO:1,ORDINAL7:94;
          then A321: born xL (+) born y1R in born x (+) born y1
          by A318,ORDINAL1:10;
          A322: born xL (+) born y1R in D by A320,A318,ORDINAL1:10,A312;
          reconsider xLy1R=xL*y1R as Surreal by A321,A2,A312;
          (xLy1) +' xy1R +' -' xLy1R in comp(L_x,x,y1,R_y1)
          by A311,A309,A313,Def14;
          then (xLy1) +' xy1R +' -' xLy1R in R_xy1 by A315,XBOOLE_0:def 3;
          then A323: xy1 < xLy1 + xy1R - xLy1R by A316;
          then A324: xy1 + xLy1R < xLy1 + xy1R by Th42;
          A325: born y1 (+) born y2 (+) born x = born x (+) born y1 (+) born y2
          by ORDINAL7:68;
          A326: born y1R (+) born x c= D by A318,A312,ORDINAL1:def 2;
          A327: y1R*x = xy1R & y2*x = xy2 by A318,A312,ORDINAL1:def 2,A313,A3;
          A328:  born y1R (+) born xL c= D
          by A320,A318,ORDINAL1:10,A312,ORDINAL1:def 2;
          A329: y1R*xL  = xLy1R by A3,A322,ORDINAL1:def 2;
          A330: y2*xL =xLy2 by A312,A313,A3;
          A331: born y1R (+) born y2 in born y1 (+) born y2
          by A317,SURREALO:1,ORDINAL7:94;
          then A332: born y1R (+) born y2 (+) born x in
          born y1 (+) born y2 (+) born x by ORDINAL7:94;
          then A333: born y1R (+) born y2 (+) born x in E by A241,A325;
          born y1R (+) born y2 (+) born xL in
          born y1 (+) born y2 (+) born xL &
          born y1 (+) born y2 (+) born xL in
          born y1 (+) born y2 (+) born x by A319,SURREALO:1,A331,ORDINAL7:94;
          then A334:born y1R (+) born y2 (+) born xL in
          born y1 (+) born y2 (+) born x by ORDINAL1:10;
          per cases by A309;
          suppose A335:y1R < y2;
            born x (+) born y1R (+) born y2 in E by A333,ORDINAL7:68;
            then A336: xLy2+xy1R < xLy1R+xy2
            by A159,A311,A314,A335,A328,A312,A313,A326;
            xy1 + xLy1R <= xLy1 + xy1R by A323,Th42;
            then A337: xy1 + xLy1R + (xLy2+xy1R) < xLy1 + xy1R+(xLy1R+xy2)
            by A336,Th44;
            A338: xy1 + xLy1R + (xLy2+xy1R) = xy1 + (xLy1R + (xLy2+xy1R))
            by Th37
            .=xy1 + (xLy2+(xLy1R+xy1R)) by Th37
            .=xy1 + xLy2+(xLy1R+xy1R) by Th37;
            xLy1 + xy1R+(xLy1R+xy2) = xLy1 + (xy1R+(xLy1R+xy2)) by Th37
            .= xLy1 + ((xy1R+xLy1R)+xy2) by Th37
            .= xLy1 + xy2+(xy1R+xLy1R) by Th37;
            hence xLy2+ xy1 < xLy1+xy2 by A337,A338,Th43;
          end;
          suppose A339: y1R == y2;
            A340: xy1R == xy2 by A332,A241,A325,A159,A312,A326,A339,A327;
            A341: xLy1R == xLy2
            by A328,A330,A329,A312,A339,A334,A159,A241,A325;
            A342: xy1 + xLy2 <= xy1 + xLy1R by A341,Th32;
            A343:xLy1 + xy1R <= xLy1 + xy2 by A340,Th32;
            xy1 + xLy2 < xLy1 + xy1R by A324,A342,SURREALO:4;
            hence xLy2+ xy1 < xLy1+xy2 by A343,SURREALO:4;
          end;
        end;
        for xR be Surreal st xR in R_x holds P3[D,x,xR,y1,y2]
        proof
          let xR be Surreal such that A344: xR in R_x;
          let xy2,xRy1,xy1,xRy2 be Surreal such that
          A345:born x (+) born y1 c= D & born xR (+) born y1 c= D &
          born x (+) born y2 c= D & born xR (+) born y2 c= D and
          A346: xy1 = x*y1 & xy2=x*y2 & xRy1=xR*y1 & xRy2=xR*y2 and
          A347: x < xR & y1 < y2;
          A348: xy1 = [comp(L_x,x,y1,L_y1) \/ comp(R_x,x,y1,R_y1),
          comp(L_x,x,y1,R_y1) \/ comp(R_x,x,y1,L_y1)] by A346,Th50;
          A349: L_xy1  << {xy1} << R_xy1 & xy1 in {xy1}
          by SURREALO:11,TARSKI:def 1;
          A350: y1R in L_y1 \/ R_y1 by A309,XBOOLE_0:def 3;
          A351: born x (+) born y1R in born x (+) born y1
          by SURREALO:1,A350,ORDINAL7:94;
          then reconsider xy1R=x*y1R as Surreal by A2,A345;
          A352: xR in L_x \/ R_x by A344,XBOOLE_0:def 3;
          A353: born xR (+) born y1R in born x (+) born y1R
          by A352,SURREALO:1,ORDINAL7:94;
          then A354: born xR (+) born y1R in D by A345,A351,ORDINAL1:10;
          then reconsider xRy1R=xR*y1R as Surreal by A2;
          (xRy1) +' xy1R +' -' xRy1R in comp(R_x,x,y1,R_y1)
          by A344,A309,A346,Def14;
          then (xRy1) +' xy1R +' -' xRy1R in L_xy1 by A348,XBOOLE_0:def 3;
          then A355: xRy1 + xy1R - xRy1R <xy1 by A349;
          then A356: xRy1 + xy1R < xy1 + xRy1R by Th41;
          A357: born y1 (+) born y2 (+) born x = born x (+) born y1 (+) born y2
          by ORDINAL7:68;
          A358: born y1R (+) born x c= D by A351,A345,ORDINAL1:def 2;
          A359: y1R*x = xy1R & y2*x = xy2
          by A345,A346,A3,A351,ORDINAL1:def 2;
          A360:  born y1R (+) born xR c= D by A353,A345,A351,ORDINAL1:10,
          ORDINAL1:def 2;
          A361: y1R*xR  = xRy1R by A3,A354,ORDINAL1:def 2;
          A362: y2*xR =xRy2 by A345,A346,A3;
          A363: born y1R (+) born y2 in born y1 (+) born y2
          by A350,SURREALO:1,ORDINAL7:94;
          then A364: born y1R (+) born y2 (+) born x in
          born y1 (+) born y2 (+) born x by ORDINAL7:94;
          then A365: born y1R (+) born y2 (+) born x in E by A241,A357;
          born y1R (+) born y2 (+) born xR in
          born y1 (+) born y2 (+) born xR &
          born y1 (+) born y2 (+) born xR in
          born y1 (+) born y2 (+) born x by A352,SURREALO:1,A363,ORDINAL7:94;
          then
          A366: born y1R (+) born y2 (+) born xR in
          born y1 (+) born y2 (+) born x by ORDINAL1:10;
          per cases by A309;
          suppose A367:y1R < y2;
            born x (+) born y1R (+) born y2 in E by A365,ORDINAL7:68;
            then A368: xy2+xRy1R < xy1R+xRy2
            by A344,A159,A347,A367,A360,A345,A346,A358;
            xRy1 + xy1R <= xy1 + xRy1R by A355,Th41;
            then A369: xRy1 + xy1R +(xy2+xRy1R) < xy1 + xRy1R +(xy1R+xRy2)
            by A368,Th44;
            A370: xRy1 + xy1R +(xy2+xRy1R) = xRy1 + (xy1R +(xRy1R+xy2))
            by Th37
            .= xRy1 + ((xy1R +xRy1R)+xy2) by Th37
            .=(xy1R + xRy1R)+ (xy2+xRy1) by Th37;
            xy1 + xRy1R +(xy1R+xRy2) = xy1 + (xRy1R +(xy1R+xRy2)) by Th37
            .= xy1 + ((xRy1R +xy1R)+xRy2) by Th37
            .= xy1 +xRy2 +(xRy1R +xy1R) by Th37;
            hence xy2+ xRy1 < xy1+xRy2 by A370,A369,Th43;
          end;
          suppose A371: y1R == y2;
            A372: xy1R == xy2 by A159,A364,A241,A357,A345,A358,A371,A359;
            A373: xRy1R == xRy2
            by A360,A362,A361,A345,A371,A366,A159,A241,A357;
            A374: xy1+ xRy1R <= xy1+ xRy2 by A373,Th32;
            A375: xy2 + xRy1 <= xy1R + xRy1 by A372,Th32;
            xRy1 + xy1R < xy1 + xRy2 by A356,A374,SURREALO:4;
            hence xy2+ xRy1 < xy1+xRy2 by A375,SURREALO:4;
          end;
        end;
        hence thesis by A310;
      end;
    end;
    A376:for E be Ordinal holds Q24[E] from ORDINAL1:sch 2(A158);
    thus for x1,x2,y be Surreal holds P2[D,x1,x2,y]
    proof
      let x1,x2,y be Surreal;
      born x1 (+) born x2 (+) born y c= born x1 (+) born x2 (+) born y;
      hence P2[D,x1,x2,y] by A376;
    end;
    defpred QEE[Ordinal] means
    for x1,x2,y1,y2 be Surreal st born x1 (+) born x2 c= $1
    holds P3[D,x1,x2,y1,y2];
    A377: for E be Ordinal st for C be Ordinal st C in E holds QEE[C]
    holds QEE[E]
    proof
      let E be Ordinal such that A378:for C be Ordinal st C in E holds QEE[C];
      let x1,x2,y1,y2 be Surreal such that
      A379: born x1 (+) born x2 c= E;
      let x1y2,x2y1,x1y1,x2y2 be Surreal such that
      A380: born x1 (+) born y1 c= D & born x2 (+) born y1 c= D &
      born x1 (+) born y2 c= D & born x2 (+) born y2 c= D and
      A381:x1y1=x1*y1 & x1y2=x1*y2 & x2y1=x2*y1 & x2y2=x2*y2 &
      x1 < x2 & y1 < y2;
      per cases by A381,SURREALO:13;
      suppose ex x1R be Surreal st x1R in R_x1 & x1 < x1R <= x2;
        then consider x1R be Surreal such that
        A382: x1R in R_x1 & x1 < x1R <= x2;
        x1R in L_x1 \/ R_x1 by A382,XBOOLE_0:def 3;
        then A383: born x1R (+) born x2 in born x1 (+) born x2 &
        born x1R (+) born y1 in born x1 (+) born y1 &
        born x1R (+) born y2 in born x1 (+) born y2 by SURREALO:1,ORDINAL7:94;
        then reconsider x1Ry1=x1R*y1,
        x1Ry2=x1R*y2
        as Surreal by A2,A380;
        A384: born x1R (+) born y1 c= D & born x1R (+) born y2 c= D
        by A383,A380,ORDINAL1:def 2;
        per cases by A382;
        suppose A385: x1R == x2;
          born x1R (+) born x2 (+) born y1 c=born x1R (+) born x2 (+) born y1;
          then A386:x1Ry1 == x2y1 by A376,A385,A384,A380,A381;
          born x1R (+) born x2 (+) born y2 c=born x1R (+) born x2 (+) born y2;
          then A387: x1Ry2 == x2y2 by A376,A385,A384,A380,A381;
          born x1 (+) born y1 (+) born y2 c=born x1 (+) born y1 (+) born y2;
          then A388: x1y2 + x1Ry1 < x1y1 + x1Ry2 by A376,A382,A380,A381,A384;
          A389:x1y1 + x1Ry2 <= x1y1+x2y2 by A387,Th32;
          x1y2+ x2y1 <= x1y2+x1Ry1 by A386,Th32;
          then x1y2+ x2y1 < x1y1 + x1Ry2 by A388,SURREALO:4;
          hence  x1y2+x2y1 < x1y1+x2y2 by SURREALO:4,A389;
        end;
        suppose  x1R < x2;
          then  A390: x1Ry2+x2y1 < x1Ry1+x2y2 by A384,A383,A378,A379,A380,A381;
          born x1 (+) born y1 (+) born y2 c= born x1 (+) born y1 (+) born y2;
          then x1y2+x1Ry1 <= x1y1 +x1Ry2 by A376,A384,A382,A380,A381;
          then A391: x1Ry2+x2y1 +(x1y2+x1Ry1) < x1Ry1+x2y2 +(x1y1 +x1Ry2)
          by A390,Th44;
          A392: x1Ry2+x2y1 +(x1y2+x1Ry1) = x1Ry2+(x2y1 +(x1y2+x1Ry1)) by Th37
          .= x1Ry2+((x2y1 +x1y2)+x1Ry1) by Th37
          .= x1Ry2+x1Ry1+ (x1y2+x2y1) by Th37;
          x1Ry1+x2y2 +(x1y1 +x1Ry2) = x1Ry1+(x2y2 +(x1y1 +x1Ry2)) by Th37
          .= x1Ry1+((x2y2 +x1y1) +x1Ry2) by Th37
          .= x1Ry1 +x1Ry2 +(x2y2 +x1y1) by Th37;
          hence x1y2+x2y1 < x1y1+x2y2 by A391,A392,Th43;
        end;
      end;
      suppose ex x2L be Surreal st x2L in L_x2 & x1 <= x2L < x2;
        then consider x2L be Surreal such that
        A393: x2L in L_x2 & x1 <= x2L < x2;
        x2L in L_x2 \/ R_x2 by A393,XBOOLE_0:def 3;
        then A394: born x2L (+) born x1 in born x1 (+) born x2 &
        born x2L (+) born y1 in born x2 (+) born y1 &
        born x2L (+) born y2 in born x2 (+) born y2 by SURREALO:1,ORDINAL7:94;
        then reconsider x2Ly1=x2L*y1,x2Ly2=x2L*y2 as Surreal by A2,A380;
        A395: born x2L (+) born y1 c= D & born x2L (+) born y2 c= D
        by A394,A380,ORDINAL1:def 2;
        per cases by A393;
        suppose A396: x1 == x2L;
          born x1 (+) born x2L (+) born y1 c=born x1 (+) born x2L (+) born y1;
          then A397:x1y1 == x2Ly1 by A376,A396,A395,A380,A381;
          born x1 (+) born x2L (+) born y2 c=born x1 (+) born x2L (+) born y2;
          then A398: x1y2 == x2Ly2 by A376,A396,A395,A380,A381;
          born x2 (+) born y1 (+) born y2 c=born x2 (+) born y1 (+) born y2;
          then A399: x2Ly2 + x2y1 < x2Ly1 + x2y2 by A376,A393,A380,A381,A395;
          A400: x2Ly1 + x2y2 <=x1y1+x2y2 by A397,Th32;
          x1y2+x2y1 <= x2Ly2 + x2y1 by A398,Th32;
          then x1y2+x2y1 < x2Ly1 + x2y2 by A399,SURREALO:4;
          hence  x1y2+x2y1 < x1y1+x2y2 by A400,SURREALO:4;
        end;
        suppose x1 < x2L;
          then  A401: x1y2+x2Ly1 < x1y1+x2Ly2 by A395,A394,A378,A379,A380,A381;
          born x2 (+) born y1 (+) born y2 c= born x2 (+) born y1 (+) born y2;
          then x2Ly2+x2y1 <= x2Ly1 +x2y2 by A376,A395,A393,A380,A381;
          then A402: x2Ly2+x2y1+(x1y2+x2Ly1) < x2Ly1 +x2y2 +(x1y1+x2Ly2)
          by A401,Th44;
          A403: x2Ly2+x2y1+(x1y2+x2Ly1) = x2Ly2+(x2y1+(x1y2+x2Ly1)) by Th37
          .= x2Ly2+((x2y1+x1y2)+x2Ly1) by Th37
          .= (x1y2 + x2y1)+(x2Ly2+x2Ly1) by Th37;
          x2Ly1 +x2y2 +(x1y1+x2Ly2) = x2Ly1 +(x2y2 +(x1y1+x2Ly2)) by Th37
          .= x2Ly1 +((x2y2 +x1y1)+x2Ly2) by Th37
          .= (x1y1+x2y2) + (x2Ly2+x2Ly1) by Th37;
          hence  x1y2+x2y1 < x1y1+x2y2 by A402,A403,Th43;
        end;
      end;
    end;
    A404:for E be Ordinal holds QEE[E] from ORDINAL1:sch 2(A377);
    let x1,x2,y1,y2 be Surreal;
    born x1 (+) born x2 c= born x1 (+) born x2;
    hence thesis by A404;
  end;
  A405:for E be Ordinal holds Q[E] from ORDINAL1:sch 2(A1);
  thus for x,y holds x*y is Surreal by A405;
  thus for x,y holds x*y = y*x by A405;
  thus for x1,x2,y,x1y,x2y be Surreal st x1 == x2 & x1y = x1*y & x2y = x2*y
  holds  x1y == x2y
  proof
    let x1,x2,y,x1y,x2y be Surreal such that
    A406:x1 == x2 & x1y = x1*y & x2y = x2*y;
    set B1 = born x1 (+) born y,B2 = born x2 (+) born y,B=B1\/B2;
    B1 c= B & B2 c= B by XBOOLE_1:7;
    hence thesis by A406,A405;
  end;
  let x1,x2,y1,y2,x1y2,x2y1,x1y1,x2y2 be Surreal such that
  A407:x1y1=x1*y1 & x1y2=x1*y2 & x2y1=x2*y1 & x2y2=x2*y2 & x1 < x2 & y1 < y2;
  set B11 = born x1 (+) born y1,
  B12 = born x1 (+) born y2,
  B21 = born x2 (+) born y1,
  B22 = born x2 (+) born y2,
  B1 = B11\/B12, B2 = B21\/B22,B=B1\/B2;
  B11 c= B1 & B12 c= B1 & B21 c= B2 & B22 c= B2 &
  B1 c= B1\/B2 & B2 c= B1\/B2 by XBOOLE_1:7;
  then B11 c= B1\/B2 & B12 c= B1\/B2 & B21 c= B1\/B2 & B22 c= B1\/B2
  by XBOOLE_1:1;
  hence x1y2+x2y1 < x1y1+x2y2 by A405,A407;
end;
