theorem Th64:
  r in p & f = Swap(id dom T,p,q) implies
  (T,p,q)incl.(r,q) = [f.r,f.q] & (T,p,q)incl.(r,p) = [f.r,f.p]
  proof assume that
A1: r in p and
A2: f = Swap(id dom T,p,q);
    set X = dom T;
    set i = id X;
    set s = Swap(i,p,q);
    set h = [:s,s:];
    set Y = (succ q)\p;
    set Z1 = [:{p},Y:], Z2 = [:Y,{q}:];
    set g = id(Z1\/Z2);
    dom i = X; then
A3: dom s = X by FUNCT_7:99;
    not p c= r by A1,ORDINAL6:4; then
    r nin {p} & r nin Y by Th59,TARSKI:def 1; then
    [r,p] nin Z1 & [r,p] nin Z2 & [r,q] nin Z1 & [r,q] nin Z2
    by ZFMISC_1:87; then
A4: [r,q] nin dom g & [r,p] nin dom g by XBOOLE_0:def 3;
    hence (T,p,q)incl.(r,q) = h.(r,q) by FUNCT_4:11
    .= [f.r,f.q] by A2,A3,FUNCT_3:def 8;
    thus (T,p,q)incl.(r,p) = h.(r,p) by A4,FUNCT_4:11
    .= [f.r,f.p] by A2,A3,FUNCT_3:def 8;
  end;
