reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];
reserve R for Relation of A,B;
reserve S for Relation of B,C;

theorem :: (13.1)
  proj1 (R*S) = (R~).:(proj1 S) & proj1 (R*S) c= proj1 R
proof
  thus
A1: proj1 (R*S) = (R*S)~.:C by Th50
    .= (S~*R~).:C by RELAT_1:35
    .= R~.:(S~.:C) by RELAT_1:126
    .= R~.:(proj1 S) by Th50;
  proj1 S = dom S;
  then proj1 (R*S) c= R~.:B by A1,RELAT_1:123;
  hence thesis by Th50;
end;
