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 Th63:
  r <> p & s <> q & f = Swap(id dom T,p,q)
  implies (T,p,q)incl.(r,s) = [f.r,f.s]
  proof assume that
A1: r <> p & s <> q and
A2: f = Swap(id dom T,p,q);
    set Y = (succ q)\p;
    set Z1 = [:{p},Y:], Z2 = [:Y,{q}:];
    set g = id(Z1\/Z2);
A3: dom f = dom id dom T by A2,FUNCT_7:99 .= dom T;
    r nin {p} & s nin {q} by A1,TARSKI:def 1; then
    [r,s] nin Z1 & [r,s] nin Z2 by ZFMISC_1:87; then
    [r,s] nin dom g by XBOOLE_0:def 3;
    hence (T,p,q)incl.(r,s) = [:f,f:].(r,s) by A2,FUNCT_4:11
    .= [f.r,f.s] by A3,FUNCT_3:def 8;
  end;
