reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;
reserve O for connected non empty Poset;
reserve R,Q for array of O;
reserve T for non empty array of O;
reserve p,q,r,s for Element of dom T;

theorem Th61:
  dom (T,p,q)incl = [:dom T, dom T:] & rng (T,p,q)incl c= [:dom T, dom T:]
  proof
    set X = dom T;
    set i = id X;
    set s = Swap(i,p,q);
    set f = [:s,s:];
    set Y = (succ q)\p;
    set Z1 = [:{p},Y:], Z2 = [:Y,{q}:];
    set g = id(Z1\/Z2);
    dom i = X; then
A1: dom s = X by FUNCT_7:99;
A2: {p} c= X & {q} c= X & Y c= X by Th60,ZFMISC_1:31;
A3: [:X, X:] \/ (Z1\/Z2) = [:X, X:] \/ Z1 \/Z2 by XBOOLE_1:4
    .= [:X, X:] \/Z2 by A2,XBOOLE_1:12,ZFMISC_1:96
    .= [:X,X:] by A2,XBOOLE_1:12,ZFMISC_1:96;
    thus dom (T,p,q)incl = dom f \/ dom g by FUNCT_4:def 1
    .= dom f \/ (Z1\/Z2)
    .= [:X, X:] by A1,A3,FUNCT_3:def 8;
A4: rng g = Z1\/Z2 & rng i = X;
    rng s = X by A4,FUNCT_7:103; then
    rng f = [:X,X:] by FUNCT_3:67;
    hence rng (T,p,q)incl c= [:X,X:] by A3,A4,FUNCT_4:17;
  end;
