set B = the Sorts of U1 (\/) the Sorts of U2;
the Sorts of U2 is MSSubset of U0 by Def9;
then A2: the Sorts of U2 c= the Sorts of U0 by PBOOLE:def 18;
the Sorts of U1 is MSSubset of U0 by Def9;
then the Sorts of U1 c= the Sorts of U0 by PBOOLE:def 18;
then the Sorts of U1 (\/) the Sorts of U2 c= the Sorts of U0 by A2, PBOOLE:16;
then reconsider B = the Sorts of U1 (\/) the Sorts of U2 as MSSubset of U0 by PBOOLE:def 18;
let W1, W2 be strict MSSubAlgebra of U0; :: thesis: ( ( for A being MSSubset of U0 st A = the Sorts of U1 (\/) the Sorts of U2 holds
W1 = GenMSAlg A ) & ( for A being MSSubset of U0 st A = the Sorts of U1 (\/) the Sorts of U2 holds
W2 = GenMSAlg A ) implies W1 = W2 )
assume that
A3: for A being MSSubset of U0 st A = the Sorts of U1 (\/) the Sorts of U2 holds
W1 = GenMSAlg A and
A4: for A being MSSubset of U0 st A = the Sorts of U1 (\/) the Sorts of U2 holds
W2 = GenMSAlg A ; :: thesis: W1 = W2
W1 = GenMSAlg B by A3;
hence W1 = W2 by A4; :: thesis: verum