 reserve W for WA-Lattice;
 reserve a,b,c for Element of W;
 reserve W for pcs-Compatible pcs-tol-reflexive pcs-tol-symmetric WAP-Lattice;
 reserve a,b for Element of W;

theorem :: Theorem 1
  for W being pcs-Compatible pcs-tol-reflexive
          pcs-tol-symmetric WAP-Lattice
  for a,b being Element of W holds
    a (--) b
     implies  :: (--) tolerates
      for x,y being Element of W st
      x in Segment (a "/\" b, a "\/" b) &
      y in Segment (a "/\" b, a "\/" b) holds
         x (--) y
  proof
    let W be pcs-Compatible
       pcs-tol-reflexive pcs-tol-symmetric WAP-Lattice;
    let a,b be Element of W;
    assume
A1: a (--) b;
    let x,y be Element of W;
    assume
A2: x in Segment (a "/\" b, a "\/" b) &
    y in Segment (a "/\" b, a "\/" b); then
    consider x2 being Element of W such that
A3: x = x2 &
    (a "/\" b <= x2 <= a "\/" b or a "\/" b <= x2 <= a "/\" b);
AA3:a "/\" b <= x <= a "\/" b or a "\/" b <= x <= a "/\" b by A3;
    consider y2 being Element of W such that
BA3: y = y2 &
    (a "/\" b <= y2 <= a "\/" b or a "\/" b <= y2 <= a "/\" b) by A2;
AA4:  a "/\" b <= y <= a "\/" b or a "\/" b <= y <= a "/\" b by BA3;
C1:   b "/\" b = b by LemmaId;
W0:   x (--) x & y (--) y by LemmaRefl;
      b (--) b by LemmaRefl; then
W1:   (a "/\" b) (--) b by C1,A1,CompDef;
      b (--) a & a (--) a by A1,LemmaRefl; then
      (a "/\" b) (--) (a "/\" a) by CompDef; then
      (a "/\" b) (--) a by LemmaId; then
      (a "/\" b) "\/" (a "/\" b) (--) a "\/" b by CompDef,W1; then
YY:   (a "/\" b) (--) a "\/" b by LemmaId2;

    per cases by AA3,AA4;

    suppose
XX1:  a "/\" b <= x <= a "\/" b & a "/\" b <= y <= a "\/" b; then
U9:   x "/\" (a "\/" b) = x by Lemat01;
U8:   (a "\/" b) "/\" y = y by XX1,Lemat01;
U4:   (a "/\" b) (--) a "\/" b by YY; then
U1:   ((a "/\" b) "\/" x) (--) ((a "\/" b) "\/" x) by W0,CompDef;
      (a "/\" b) "\/" x = x by XX1,Lemat0; then
UU:   x (--) (a "\/" b) by U1,XX1,Lemat0;
W1:   ((a "/\" b) "\/" y) (--) ((a "\/" b) "\/" y) by U4,W0,CompDef;
      (a "/\" b) "\/" y = y by XX1,Lemat0; then
      y (--) (a "\/" b) by W1,XX1,Lemat0;
      hence thesis by U8,U9,UU,CompDef;
    end;

    suppose
XX1:  a "\/" b <= x <= a "/\" b & a "/\" b <= y <= a "\/" b; then
Y2:   (a "/\" b) "\/" x = a "/\" b by Lemat0;
      a "/\" b (--) a "\/" b by YY; then
Y1:   (a "/\" b) "\/" x (--) (a "\/" b) "\/" x by CompDef,W0;
s1:   a "/\" b (--) x by Y2,Y1,XX1,Lemat0;
S1:   (a "/\" b) "\/" y = y by XX1,Lemat0;
Y3:   (a "\/" b) "\/" y = a "\/" b by XX1,Lemat0;
      (a "/\" b) "\/" y (--) (a "\/" b) "\/" y by CompDef,W0,YY; then
s2:   y (--) (a "\/" b) by S1,Y3;
      x = x "\/" (a "\/" b) by XX1,Lemat0;
      hence thesis by S1,s1,s2,CompDef;
    end;

    suppose
XX1:  a "\/" b <= x <= a "/\" b & a "\/" b <= y <= a "/\" b; then
U2:   (a "\/" b) "\/" x = x by Lemat0;
U4:   (a "/\" b) (--) a "\/" b by YY; then
U1:   ((a "/\" b) "\/" x) (--) ((a "\/" b) "\/" x) by W0,CompDef;
u1:   ((a "/\" b) "\/" y) (--) ((a "\/" b) "\/" y) by W0,CompDef,U4;
      (a "/\" b) "\/" x = a "/\" b by XX1,Lemat0; then
UU:   x (--) (a "/\" b) by U1,U2;
U2:   (a "\/" b) "\/" y = y by XX1,Lemat0;
      (a "/\" b) "\/" y = a "/\" b by XX1,Lemat0; then
Uu:   y (--) (a "/\" b) by u1,U2;
U9:   x "/\" (a "/\" b) = x by XX1,Lemat01;
      (a "/\" b) "/\" y = y by XX1,Lemat01;
      hence thesis by Uu,U9,UU,CompDef;
    end;

    suppose
XX1:  a "/\" b <= x <= a "\/" b & a "\/" b <= y <= a "/\" b; then
S1:   (a "/\" b) "\/" x = x by Lemat0;
      a "/\" b (--) a "\/" b by YY; then
Y1:   (a "/\" b) "\/" y (--) (a "\/" b) "\/" y by CompDef,W0;
      (a "/\" b) "\/" y = a "/\" b by XX1,Lemat0; then
s1:   a "/\" b (--) y by Y1,XX1,Lemat0;
Y3:   (a "\/" b) "\/" x = a "\/" b by XX1,Lemat0;
      (a "/\" b) "\/" x (--) (a "\/" b) "\/" x by CompDef,W0,YY; then
s2:   x (--) (a "\/" b) by S1,Y3;
      y = y "\/" (a "\/" b) by XX1,Lemat0;
      hence thesis by S1,s1,s2,CompDef;
    end;
  end;
