reserve x,y,z,a,b,c,X,A for set;
reserve C,D for Coherence_Space;
reserve T for Tolerance of union C;

theorem
  C = bool union C implies Web(C) = Total union C
proof
  reconsider t = Total union C as Tolerance of union C;
  assume
A1: C = bool union C;
  now
    let x,y be object;
    thus [x,y] in t implies {x,y} in C
    proof
      assume [x,y] in t;
      then
A2:   x in union C & y in union C by ZFMISC_1:87;
      {x,y} c= union C
      by A2,TARSKI:def 2;
      hence thesis by A1;
    end;
    assume {x,y} in C;
    then x in union C & y in union C by A1,ZFMISC_1:32;
    hence [x,y] in t by TOLER_1:2;
  end;
  hence thesis by Th5;
end;
