let G be Group; :: thesis: for g1, g2 being Element of G

for H being Subgroup of G st g1 in carr H & g2 in carr H holds

g1 * g2 in carr H

let g1, g2 be Element of G; :: thesis: for H being Subgroup of G st g1 in carr H & g2 in carr H holds

g1 * g2 in carr H

let H be Subgroup of G; :: thesis: ( g1 in carr H & g2 in carr H implies g1 * g2 in carr H )

assume ( g1 in carr H & g2 in carr H ) ; :: thesis: g1 * g2 in carr H

then ( g1 in H & g2 in H ) ;

then g1 * g2 in H by Th50;

hence g1 * g2 in carr H ; :: thesis: verum

for H being Subgroup of G st g1 in carr H & g2 in carr H holds

g1 * g2 in carr H

let g1, g2 be Element of G; :: thesis: for H being Subgroup of G st g1 in carr H & g2 in carr H holds

g1 * g2 in carr H

let H be Subgroup of G; :: thesis: ( g1 in carr H & g2 in carr H implies g1 * g2 in carr H )

assume ( g1 in carr H & g2 in carr H ) ; :: thesis: g1 * g2 in carr H

then ( g1 in H & g2 in H ) ;

then g1 * g2 in H by Th50;

hence g1 * g2 in carr H ; :: thesis: verum