reserve x,y,z,a,b,c,X,A for set;
reserve C,D for Coherence_Space;
reserve T for Tolerance of union C;
reserve E for Tolerance of X;
reserve C,C1,C2 for Element of CSp(X);
reserve g for Element of FuncsC(X);
reserve l,l1,l2,l3 for Element of MapsC(X);
reserve T,T1,T2 for Element of TOL(X);
reserve f for Element of FuncsT(X);
reserve m,m1,m2,m3 for Element of MapsT(X);

theorem Th43:
  dom m2 = cod m1 implies (m2*m1)`2 = m2`2*m1`2 & dom(m2*m1) = dom
  m1 & cod(m2*m1) = cod m2
proof
  assume dom m2 = cod m1;
  then [[dom m1,cod m2],m2`2*m1`2] = m2*m1 by Def23
    .= [[dom(m2*m1),cod(m2*m1)],(m2*m1)`2] by Th41;
  hence thesis by Lm2,XTUPLE_0:1;
end;
